Overview of Magma V2.18 Features

Home , Magma (algebra)

MAGMA COMPUTER • ALGEBRA

Computational Algebra Group

University of Sydney

December 16, 2011

Magma Overview V2.18 (December 16, 2011) i

Contents

1 Introduction 1 1.1 The Magma Philosophy ...... 1 1.2 Summary of this Document ...... 1

2 The Magma Language and System 3 2.1 The Magma User Language ...... 3 2.2 The Magma Environment ...... 3

3 Groups 4 3.1 Permutation Groups ...... 4 3.1.1 Construction ...... 4 3.1.2 Base and Strong Generating Set ...... 4 3.1.3 Elementary Properties ...... 5 3.1.4 Conjugacy Classes ...... 5 3.1.5 Subgroup Constructions ...... 5 3.1.6 Actions ...... 5 3.1.7 Analysis of a Primitive Group ...... 6 3.1.8 Normal Structure ...... 6 3.1.9 Standard Quotients ...... 6 3.1.10 Subgroup Structure ...... 7 3.1.11 Automorphisms ...... 7 3.1.12 Cohomology and Representations ...... 7 3.1.13 Databases ...... 7 3.2 Matrix Groups ...... 8 3.2.1 Construction ...... 8 3.2.2 Arithmetic with Elements ...... 8 3.2.3 Actions ...... 8 3.2.4 Base and Strong Generating Set ...... 9 3.2.5 Elementary Properties ...... 9 3.2.6 Conjugacy Classes ...... 9 3.2.7 Subgroup Constructions ...... 9 3.2.8 Normal Structure ...... 9 3.2.9 Standard Quotients ...... 10 3.2.10 Automorphisms ...... 10 3.2.11 Cohomology and Representations ...... 10 3.2.12 Aschbacher Analysis ...... 10 3.2.13 Databases of Matrix Groups ...... 11 3.3 Constructive Recognition ...... 11 3.4 Finitely Presented Groups ...... 11 3.4.1 Free Groups ...... 11 3.4.2 Construction ...... 11 3.4.3 Arithmetic on Elements ...... 12 3.4.4 Basic Properties ...... 12 3.4.5 Quotients ...... 12 3.4.6 Constructing a Subgroup ...... 12 3.4.7 Operations on Subgroups of Finite Index ...... 13 3.4.8 Enumeration of Subgroups ...... 13 3.4.9 Simplifying a Presentation ...... 13 3.4.10 Actions ...... 13 3.4.11 Representation Theory ...... 13 3.4.12 Databases of Finitely Presented Groups ...... 13 3.5 Generic Abelian Groups ...... 14 ii Magma Overview V2.18 (December 16, 2011)

3.6 Finitely-Presented Abelian Groups ...... 14 3.7 Polycyclic Groups ...... 15 3.7.1 Polycyclic Groups: Construction and Arithmetic ...... 15 3.7.2 Polycyclic Groups: Basic Invariants ...... 15 3.7.3 Polycyclic Groups: Subgroup Constructions ...... 15 3.7.4 Polycyclic Groups: Normal Structure ...... 15 3.8 Finite Soluble Groups ...... 16 3.8.1 Construction ...... 16 3.8.2 Conjugacy Classes ...... 16 3.8.3 Subgroup Constructions ...... 16 3.8.4 Normal Structure ...... 17 3.8.5 Subgroup Structure ...... 17 3.8.6 Automorphisms and Representations ...... 17 3.9 Finite p-Groups ...... 17 3.9.1 Construction ...... 17 3.9.2 Normal Structure ...... 18 3.9.3 Isomorphisms and Automorphism Groups ...... 18 3.9.4 Character Theory ...... 18 3.10 Groups Deﬁned by Rewrite Systems ...... 18 3.11 Automatic Groups ...... 18 3.12 Groups with Elements given as Straight-Line Programs ...... 19 3.13 Braid Groups ...... 19 3.13.1 Constructing and Accessing Braid Groups ...... 19 3.13.2 Constructing and Accessing Elements ...... 19 3.13.3 Normal Forms of Elements ...... 19 3.13.4 Arithmetic Operations with Elements ...... 19 3.13.5 Boolean Predicates ...... 20 3.13.6 Lattice Operations ...... 20 3.13.7 Conjugates ...... 20 3.13.8 Homomorphisms ...... 20

4 Semigroups and Monoids 21 4.1 Finitely Presented Semigroups ...... 21 4.2 Monoids Deﬁned by Rewrite Systems ...... 21

5 Rings and their Fields 22 5.1 The Rational Field and its Ring of Integers ...... 23 5.1.1 Arithmetic ...... 23 5.1.2 Residue Class Rings of Z ...... 23 5.1.3 Primality and Factorization ...... 23 5.1.4 The Number Field Sieve ...... 24 5.2 Dirichlet Characters ...... 24 5.3 Univariate Polynomial Rings ...... 25 5.3.1 Creation and Ring Operations ...... 25 5.3.2 Creation of Special Polynomials ...... 25 5.3.3 Arithmetic with Polynomials ...... 25 5.3.4 GCD and Factorization ...... 27 5.3.5 Arithmetic with Ideals ...... 27 5.4 Residue Class Rings of Univariate Polynomial Rings ...... 27 5.5 Finite Fields ...... 28 5.5.1 Construction ...... 28 5.5.2 Arithmetic ...... 28 5.5.3 Roots and Polynomial Factorization ...... 28 5.5.4 Discrete Logarithms ...... 29 Magma Overview V2.18 (December 16, 2011) iii

5.5.5 Derived Structures ...... 29

6 Global Arithmetic Fields 30 6.1 Number Fields and their Orders ...... 30 6.1.1 Number Fields ...... 30 6.1.2 Orders and Fractional Ideals ...... 30 6.1.3 Invariants ...... 31 6.1.4 Diophantine (and other) Equations ...... 31 6.1.5 Quadratic Fields ...... 31 6.1.6 Cyclotomic Fields ...... 31 6.2 Galois Theory of Number Fields ...... 32 6.3 Class Field Theory of Number Fields ...... 32 6.4 General Algebraic Function Fields ...... 33 6.4.1 Rational Function Fields ...... 33 6.4.2 Algebraic Function Fields ...... 33 6.4.3 Orders of Algebraic Function Fields ...... 34 6.4.4 Elements of Algebraic Function Fields and their Orders ...... 34 6.4.5 Ideals of Orders of Algebraic Function Fields ...... 34 6.4.6 Places of Algebraic Function Fields ...... 35 6.4.7 Divisors of Algebraic Function Fields ...... 35 6.4.8 Diﬀerentials of Algebraic Function Fields ...... 35 6.4.9 Divisor Class Groups for Global Algebraic Function Fields ...... 36 6.4.10 Class Field Theory for Algebraic Function Fields ...... 36 6.5 Algebraically Closed Fields ...... 37

7 Local Arithmetic Fields 38 7.1 Discrete Valuation Rings ...... 39 7.2 The Real and Complex Fields ...... 40 7.3 Newton Polygons ...... 40 7.4 p-adic Rings and their Extensions ...... 41 7.4.1 Construction ...... 41 7.4.2 Arithmetic ...... 41 7.4.3 Polynomial Factorization ...... 41 7.4.4 Class ﬁeld theory ...... 41 7.5 General Local Fields ...... 42 7.6 Galois Rings ...... 42 7.7 Power, Laurent and Puiseux Series Rings ...... 43 7.8 Lazy Power Series Rings ...... 43

8 Linear Algebra and Module Theory 44 8.1 Matrices ...... 44 8.1.1 Representation of Matrices ...... 44 8.1.2 Arithmetic ...... 44 8.1.3 Echelon Form and Related Operations ...... 45 8.1.4 Canonical Forms ...... 45 8.2 Sparse Matrices ...... 46 8.3 Vector Spaces ...... 46 8.3.1 Construction ...... 46 8.3.2 Construction ...... 46 8.3.3 Subspaces and Quotient Spaces ...... 46 8.3.4 Bases ...... 47 8.3.5 Homomorphisms ...... 47 8.3.6 Quadratic Forms ...... 47 8.4 Free Modules ...... 47 iv Magma Overview V2.18 (December 16, 2011)

8.4.1 Basic Operations ...... 47 8.4.2 Homomorphisms ...... 47 8.5 Modules over Dedekind domains ...... 48 8.6 Chain Complexes ...... 48

9 Lattices and Quadratic Forms 49 9.1 Integral Lattices ...... 49 9.1.1 Constructions and Operations ...... 49 9.1.2 Reduction Algorithms ...... 49 9.1.3 Properties ...... 50 9.2 Lattices with Group Action: G-Lattices ...... 51 9.2.1 Automorphism Groups ...... 51 9.2.2 G-Lattices ...... 51 9.3 Quadratic Forms ...... 52 9.3.1 General Quadratic Forms ...... 52 9.3.2 Binary Quadratic Forms ...... 52

10 Associative Algebras 53 10.1 Finitely Presented Associative Algebras ...... 53 10.2 Exterior Algebras ...... 54 10.3 General Finite-Dimensional Algebras ...... 54 10.4 Finite-Dimensional Associative Algebras ...... 54 10.4.1 Constructions and Element Operations ...... 54 10.4.2 Ideal and Subalgebra Structure ...... 54 10.4.3 Orders ...... 55 10.5 Matrix Algebras ...... 56 10.5.1 Constructions and Element Operations ...... 56 10.5.2 Ideal and Subalgebra Structure ...... 56 10.5.3 Presentations ...... 56 10.6 Group Algebras ...... 57 10.7 Quaternion Algebras ...... 57 10.8 Basic Algebras ...... 58 10.8.1 Construction ...... 58 10.8.2 Modules and Cohomology ...... 58

11 Representation Theory 59 11.1 Modules over an Algebra ...... 59 11.1.1 Creation ...... 59 11.1.2 Constructions ...... 59 11.1.3 Submodules and Quotient Modules ...... 59 11.1.4 Structure ...... 60 11.1.5 Homomorphisms ...... 60 11.2 Ordinary Representations ...... 60 11.2.1 Splitting of G-modules ...... 60 11.2.2 Construction of Irreducible G-modules ...... 60 11.2.3 Change of Ring ...... 61 11.2.4 Properties ...... 61 11.3 Representations of Symmetric Groups ...... 61 11.4 Character Theory ...... 61 Magma Overview V2.18 (December 16, 2011) v

12 Lie Theory 62 12.1 Coxeter systems ...... 62 12.2 Root Systems ...... 62 12.2.1 Creating Root Systems ...... 62 12.2.2 Operations and Properties ...... 62 12.2.3 Roots and Coroots ...... 63 12.3 Root Data ...... 63 12.3.1 Constructions ...... 63 12.3.2 Operations and Properties ...... 63 12.3.3 Roots, Coroots and Weights ...... 63 12.4 Coxeter Groups ...... 64 12.4.1 General Coxeter Groups as FP-Groups ...... 64 12.4.2 Finite Coxeter Groups as Permutation Groups ...... 64 12.5 Complex Reﬂection Groups ...... 65 12.6 Finite-Dimensional Lie Algebras ...... 65 12.6.1 Construction and Arithmetic ...... 65 12.6.2 Properties and Invariants ...... 65 12.6.3 Arithmetic of Subalgebras and Ideals ...... 65 12.6.4 Structure ...... 66 12.6.5 Representations ...... 66 12.6.6 Universal enveloping algebras ...... 66 12.6.7 Finitely Presented Lie Algebras ...... 66 12.7 Quantized Enveloping Algebras ...... 66 12.8 Groups of Lie Type ...... 66 12.8.1 Creating Groups of Lie type ...... 66 12.8.2 Operations and Properties ...... 67 12.8.3 Automorphisms ...... 67 12.8.4 Representation Theory ...... 67 12.9 Finite Groups of Lie Type ...... 67

13 Commutative Algebra 68 13.1 Multivariate Polynomial Rings ...... 69 13.1.1 Creation and Ring Operations ...... 69 13.1.2 Arithmetic ...... 69 13.1.3 GCD and Factorization ...... 69 13.1.4 Gr¨obnerBasis ...... 70 13.1.5 Arithmetic with Ideals ...... 70 13.1.6 Invariants for Ideals ...... 71 13.1.7 Gradings ...... 71 13.2 Boolean Polynomial Rings ...... 71 13.3 Aﬃne Algebras ...... 72 13.3.1 Creation and Operations ...... 72 13.3.2 Arithmetic with Ideals ...... 72 13.4 Modules over Multivariate Rings ...... 72 13.4.1 Creation and Operations ...... 72 13.4.2 Submodules ...... 73 13.4.3 Homology ...... 73 13.5 Invariants for Groups ...... 74 13.5.1 Construction of Primary and Secondary Invariants ...... 74 13.5.2 The Ring of Invariants ...... 74 13.5.3 Properties ...... 74 13.5.4 Invariants of Linear Algebraic Groups ...... 75 13.5.5 Invariant Fields ...... 75 13.5.6 Invariants of the Symmetric Group ...... 75 vi Magma Overview V2.18 (December 16, 2011)

14 Algebraic Geometry 76 14.1 Schemes ...... 76 14.1.1 Ambient Spaces ...... 76 14.1.2 Creation and Properties ...... 76 14.1.3 Mappings ...... 77 14.1.4 Automorphisms ...... 77 14.1.5 Isomorphic Projections ...... 77 14.1.6 Linear Systems ...... 77 14.2 Coherent Sheaves ...... 78 14.3 General Algebraic Curves ...... 78 14.3.1 Construction and General Properties ...... 78 14.3.2 Mappings ...... 79 14.3.3 Automorphism groups ...... 79 14.3.4 Local Analysis ...... 79 14.3.5 Ordinary Plane Curves ...... 79 14.3.6 Function Field ...... 80 14.3.7 Divisors and the Riemann-Roch Theorem ...... 80 14.3.8 Diﬀerentials ...... 80 14.3.9 Resolution Graphs and Splice Diagrams ...... 80 14.4 Algebraic Surfaces ...... 81 14.4.1 Formal Desingularisation and Classiﬁcation ...... 81 14.4.2 Parametrization Of Rational Surfaces ...... 81 14.5 Toric Varieties ...... 82 14.5.1 Toric lattices ...... 82 14.5.2 Cones and Polyhedra ...... 82 14.5.3 Fans ...... 83 14.5.4 Basic operations on toric varieties ...... 83 14.5.5 Maps of toric varieties ...... 83 14.5.6 Invariant divisors and Riemann-Roch spaces ...... 84 14.5.7 Mori theory ...... 84 14.6 Graded Rings and Geometric Databases ...... 85

15 Arithmetic Geometry 86 15.1 Rational Curves and Conics ...... 86 15.2 Elliptic Curves ...... 87 15.2.1 Construction and Properties ...... 87 15.2.2 Morphisms ...... 87 15.2.3 Features over finite fields ...... 87 15.2.4 Features over Q ...... 88 15.2.5 Rational points over Q ...... 88 15.2.6 Features over number fields ...... 89 15.2.7 Rational points over number fields ...... 89 15.2.8 Features over function fields ...... 89 15.2.9 Rational points over function fields ...... 89 15.3 Models of Genus One Curves ...... 90 15.4 Hyperelliptic Curves and Jacobians ...... 91 15.4.1 Construction and Properties ...... 91 15.4.2 Operations on curves over finite fields ...... 91 15.4.3 Operations on curves over Q and number fields ...... 91 15.4.4 Construction of Jacobians ...... 92 15.4.5 Kummer Surface of a genus 2 Jacobian ...... 92 15.4.6 Operations on Jacobians over finite fields ...... 92 15.4.7 Operations on Jacobians over Q and number fields ...... 92 15.5 L-Series ...... 93 Magma Overview V2.18 (December 16, 2011) vii

16 Modular Arithmetic Geometry 94 16.1 Modular Curves ...... 94 16.2 Congruence Subgroups of PSL(2,R) ...... 94 16.3 Modular Forms ...... 95 16.4 Modular Symbols ...... 95 16.5 Modular Abelian Varieties ...... 96 16.6 Brandt Modules ...... 96 16.7 Supersingular Divisors on Modular Curves ...... 97 16.8 Hilbert Modular Forms ...... 97 16.9 Modular Forms on Imaginary Quadratic Fields ...... 97

16.10Admissible Representations of GL2(Qp) ...... 97

17 Differential Galois Theory 99 17.1 Differential Rings and Fields ...... 99 17.2 Differential Operator Rings ...... 99

18 Geometry 100 18.1 Finite Planes ...... 100 18.2 Incidence Geometry ...... 101 18.2.1 Incidence Geometries ...... 101 18.2.2 Coset Geometries ...... 101 18.3 Cones and Polyhedra ...... 102

19 Combinatorial Theory 103 19.1 Enumerative Combinatorics ...... 103 19.2 Partitions and Young Tableaux ...... 103 19.3 Symmetric Function Algebras ...... 104 19.4 Graphs ...... 105 19.5 Multigraphs ...... 106 19.6 Networks ...... 106 19.7 Incidence Structures and Designs ...... 107 19.8 Hadamard Matrices ...... 107

20 Coding Theory 108 20.1 Linear Codes over Finite Fields ...... 108 20.1.1 Linear Codes: Creation ...... 108 20.1.2 Operations on Codewords ...... 108 20.1.3 Elementary Operations ...... 108 20.1.4 Basic Properties ...... 109 20.1.5 Weight Distribution ...... 109 20.1.6 Construction of Families ...... 109 20.1.7 Algebraic-Geometric codes ...... 109 20.1.8 Decoding Algebraic-Geometric codes ...... 110 20.1.9 Changing the Alphabet ...... 110 20.1.10 Combining Codes ...... 110 20.1.11 Bounds ...... 110 20.1.12 Best Known Codes ...... 110 20.1.13 Decoding Algorithms ...... 111 20.1.14 Automorphism Group ...... 111 20.1.15 Attacks on the McEliece Cryptosystem ...... 111 20.2 Low-Density Parity-Check Codes ...... 112 20.3 Linear Codes over Finite Rings ...... 113 20.3.1 Constructions ...... 113 20.3.2 Elementary Operations and Properties ...... 113 viii Magma Overview V2.18 (December 16, 2011)

20.3.3 Weight Distribution ...... 113 20.4 Additive Codes ...... 114 20.4.1 Constructions ...... 114 20.4.2 Operations and Properties ...... 114 20.4.3 Weight Distribution ...... 114 20.5 Quantum Error-Correcting Codes ...... 115 20.5.1 Constructions ...... 115 20.5.2 Quantum Codes: Basic Properties ...... 115

21 Cryptography 116 21.1 Pseudo-Random Sequences ...... 116

22 Mathematical Databases 117 22.1 Algebraic Geometry ...... 117 22.2 Coding Theory ...... 117 22.3 Finite Fields ...... 118 22.4 Graph Theory ...... 118 22.5 Group Theory ...... 118 22.6 Hadamard Matrices ...... 120 22.7 Lattices ...... 120 22.8 Lie Algebras ...... 120 22.9 Number Theory ...... 120 22.10Topology ...... 121

23 Documentation 122 Magma Overview V2.18 (December 16, 2011) 1

1 Introduction 1.1 The Magma Philosophy Magma is a Computer Algebra system designed to solve problems in algebra, number theory, geometry and combinatorics that may involve sophisticated mathematics and which are computa- tionally hard. Magma provides a mathematically rigorous environment which emphasizes structural computation. A key feature is the ability to construct canonical representations of structures, thereby making possible such operations as membership testing, the determination of structural properties and isomorphism testing. The kernel of Magma contains implementations of many of the important concrete classes of structure in five fundamental branches of algebra, namely group theory, ring theory, field theory, module theory and the theory of algebras. In addition, certain families of structures from algebraic geometry and finite incidence geometry are included. The main features of the Magma system include:

(a) Algebraic Design Philosophy: The design principles underpinning both the user language and system architecture are based on ideas from universal algebra and category theory. The language attempts to approximate as closely as possible the usual mathematical modes of thought and notation. In particular, the principal constructs in the user language are set, (algebraic) structure and morphism. (b) Universality: In-depth coverage of all the major branches of algebra, number theory, algebraic geometry and ﬁnite incidence geometry. (c) Integration: The facilities for each area are designed in a similar manner using generic constructors wherever possible. The uniform design makes it a simple matter to program calculations that span diﬀerent classes of mathematical structures or which involve the in- teraction of structures. (d) Performance: The intention is that Magma provide the best possible performance both in terms of the algorithms used and their implementation. The design philosophy permits the kernel implementor to choose optimal data structures at the machine level. Most of the major algorithms currently installed in the Magma kernel are state-of-the-art and give performance similar to, or better than, specialized programs.

The purpose of this document is to provide an overview of the structures and operations that are implemented in Magma. A collection of illustrative examples may be found at the Magma home page: http://magma.maths.usyd.edu.au/ – also available from this web site are a number of short papers written by experts describing applications of Magma in various branches of mathematics.

1.2 Summary of this Document Computation in Magma always takes place in one or more explicitly deﬁned structures. A family of structures whose members satisfy a common set of axioms and which share a common representation is termed a category. In the ﬁrst of the following sections we summarise the facilities of the Magma language and environment. Following this we summarise many of the kernel categories grouped together under the broad headings listed below.

• The Magma Language and System • Groups • Semigroups and Monoids • Rings and their Fields • Commutative Rings • Linear Algebra and Module Theory 2 Magma Overview V2.18 (December 16, 2011)

• Lattices and Quadratic Forms • Algebras • Representation Theory • Homological Algebra • Lie Theory • Algebraic Geometry • Finite Incidence Geometry • Diﬀerential Galois Theory • Error-correcting Codes • Cryptography • Mathematical Databases

All timings given below are for a Sun 400Mhz UltraSPARC 2 unless otherwise indicated. Magma Overview V2.18 (December 16, 2011) 3

2 The Magma Language and System 2.1 The Magma User Language – Imperative language with standard imperative-style statements and procedures – A functional subset providing closures, higher-order functions, and partial evaluation – General aggregate data types based on algebraic notions: set, sequence, mapping, magma – Universal structure constructors providing a general mechanism for the construction of magmas and mappings – Simple but powerful notation for constructing sets and sequences in a natural mathematical style – Set and sequence operations which are implemented with a strong emphasis on eﬃciency – Coercion between magmas (including automatic coercion) – A package mechanism to support modular program construction

2.2 The Magma Environment – Command completion and interactive line editing – History system with recall and editing of previous lines – A hierarchical online help facility – Packages containing user-defined intrinsics with automatic compilation – Command-line options at startup – Environment variables for configuring style of output, etc. – Get/set functions and procedures for configuring style, etc. – Verbose options for built-in functions – Logging of output, redirection of I/O – Mechanism for saving and restoring user workspaces – Special file type for fully-featured file I/O – Ability to execute system commands from within Magma – Input/output pipes for communication with external programs – UNIX commands and functions such as process ID, alarm setting, etc. – A socket mechanism for communicating with other processes. 4 Magma Overview V2.18 (December 16, 2011)

3 Groups

Group theory has been one of the Computational Algebra Group’s traditional strengths and Magma provides the user with access to nearly all of the significant algorithms for finite groups and finitely presented infinite groups (fp-groups). The important categories of group include:

• Permutation groups • Matrix groups • Finitely presented groups • Coxeter groups • Generic abelian groups • Finitely-presented abelian groups • Polycyclic groups • Finite soluble groups • Finite p-groups • Automorphism groups • Groups deﬁned by rewrite systems • Automatic groups • Groups with elements given as straight-line programs • Black-box groups • Braid groups • Congruence subgroups of PSL(2, R)

In addition, Magma provides extensive machinery for the representation theory of groups which is discussed in the Representation Theory section of this document.

3.1 Permutation Groups A permutation action may be defined on any finite set using a G-set mechanism. A huge range of permutation group algorithms (some 300) are incorporated—many of them being developed specifically for Magma. All algorithms are either deterministic or Las Vegas, that is if an answer is returned it is guaranteed correct but it is possible no answer will be returned.

3.1.1 Construction – Permutation representations for classical groups, e.g. PGL(n, q), PSp(n, q), PSU(n, q2), P Ω(n, q)

– Construction of standard groups, e.g., Sn, An. – Construction of wreath products with both types of action – Random generation of elements (Product-replacement algorithm),

3.1.2 Base and Strong Generating Set – Sims-Schreier algorithm for constructing a base and strong generating set (BSGS) – Random Schreier algorithm for constructing a BSGS (Monte Carlo algorithm) – Todd-Coxeter Schreier algorithm for constructing a BSGS – Sims variation of Schreier method for soluble groups – Fast construction of BSGS when group order is known – Brownie-Cannon-Sims algorithm for verifying a BSGS Magma Overview V2.18 (December 16, 2011) 5

The key concept for representing a permutation group is that of a base and strong generating set (BSGS). Given a BSGS for a group, its order may be deduced immediately. Brownie, Cannon and Sims (1991) showed that it is practical, in some cases at least, to construct a BSGS for short-base groups having degree up to ten million. For example, starting with six permutations of degree 8 835 156, generating the Lyons simple group, it takes Magma 5.0 hours to provably determine the order of the group they generate. The ability to construct a BSGS, coupled with the use of algorithms that make heavy use of the classiﬁcation theorem for ﬁnite simple groups, allow the determination of a great deal of structural information, such as composition factors, for short base groups of degree up to at least ten million.

3.1.3 Elementary Properties The functions listed in this section require a BSGS to be computed. – Order – Exponent – Whether abelian, nilpotent, soluble, etc. – Whether perfect, simple, etc.

3.1.4 Conjugacy Classes The conjugacy classes of elements of a group G are found using a lifting algorithm which ﬁrst ﬁnds the classes of the trivial Fitting quotient of G (algorithm of D. Holt) and then lifts these classes through the layers of an elementary abelian series for G. For example, all classes in the group 12 2 .(SL2(4) o S3) of degree 4096 and order 5, 308, 416, 000 are computed in 4 seconds. – Testing a pair of elements for conjugacy – Conjugacy classes of elements (lifting algorithm) – Power map – Class map, i.e. return the number of the class to which a given element belongs – Class matrix

3.1.5 Subgroup Constructions – Construction of subgroups, quotient groups – Normal closure, core of a subgroup – Normalizer, centralizer – Testing subgroups for conjugacy – Intersection of subgroups – Sylow p-subgroup (reduction algorithm)

3.1.6 Actions – Stabilizer of a point, set of points, sequence of points – Stabilizer of an ordered partition, conjugacy of partitions – Orbits on points, sets of points, and sequences of points – Homomorphisms induced by actions on orbits – Systems of imprimitivity (Sch¨onert-Seressalgorithm) – Homomorphisms induced by actions systems of imprimitivity – Induced actions on G-sets – Action properties: Semiregular, regular, transitive, primitive, Frobenius – Permutation representation on the cosets of a subgroup – Fast tests for the alternating/symmetric group 6 Magma Overview V2.18 (December 16, 2011)

3.1.7 Analysis of a Primitive Group – Elementary abelian regular normal subgroup (EARNS) – Action of an affine primitive group on its EARNS – Construction of the socle of a non-affine group via the O’Nan-Scott theorem – Action of a primitive group on its socle – Permutation representation of G/N, where G is a primitive group and N is its socle – O’Nan-Scott decomposition of a primitive group – Identification of a 2-transitive group

The Magma group has developed efficient methods for obtaining the O’Nan-Scott decomposition of a primitive group. The elementary abelian regular normal subgroup of an affine primitive group is constructed by a polynomial-time algorithm based on ideas published by P. Neumann (1986). For example, Magma finds the EARNS of AGL(10, 3) which has degree 59, 049 and order 17046196453240220939126401085378073952125928970649600 in 1.49 seconds. The construction of the socle and the analysis of a non-affine primitive group is performed by algorithms based on ideas of Cannon, Holt and Kantor. A 2-transitive group is identified as an abstract group using an algorithm published by Cameron and Cannon.

3.1.8 Normal Structure – Derived subgroup, derived series, soluble residual – Lower central series, nilpotent residual – p-core, Fitting subgroup, soluble radical (Unger’s polynomial-time algorithm) – Centre, upper central series – Elementary abelian series, p-central series – Socle, socle action – Chief series, chief factors, composition factors – Agemo, omega subgroups (of a p-group) – Minimal normal subgroups – Maximal normal subgroups – All normal subgroups

3.1.9 Standard Quotients – Maximal abelian quotient – Maximal soluble quotient – Socle quotient (for primitive and trivial Fitting groups) – Conjugation action on socle factors (trivial Fitting groups) – Quotient by an abelian normal subgroup – Radical quotient – Presentation on given generators (for groups of moderate order) – Presentation on strong generators – Presentation of quotient by a normal subgroup

A variety of methods are used for quotient constructions. Quotients by abelian groups use the algorithm of Luks & Seress (1997). Finitely presented quotients use a combination of the Schreier- Todd-Coxeter-Sims method of Leon (1980) and the presentation implicit in Sims’ veriﬁcation of a BSGS (see Gebhardt (2000)). Magma Overview V2.18 (December 16, 2011) 7

3.1.10 Subgroup Structure – Maximal subgroups – Frattini subgroup – Conjugacy classes of complements of a soluble normal subgroup – Conjugacy classes of subgroups, poset of subgroup classes – Conjugacy classes of subgroups satisfying a condition: cyclic, elementary abelian, abelian, nilpotent – Low index subgroups

Magma V2.14 and later contains an implementation of a very powerful algorithm for computing the maximal subgroups of a group G of moderate degree. The algorithm ﬁrst determines the maximal subgroups of the trivial Fitting quotient of G and then lifts these maximal subgroups down an elementary abelian series. Magma ﬁnds the maximal subgroups of the group of Rubik’s cube (degree 48, order 43252003274489856000) in 0.42 seconds.

3.1.11 Automorphisms – Automorphism group – Test for two permutation groups being isomorphic

The automorphism group of a permutation group G is found using a lifting algorithm which first finds the automorphisms of the trivial Fitting quotient of G by looking up the automorphism groups of any non-cyclic factors in a database. Then these automorphisms are lifted through the layers of an elementary abelian series for G. For example, the automorphism group of the group of Meffert’s puzzle (a non-soluble permutation group of degree 30 and order 283105) is found in 0.14 seconds. The group has 16 outer automorphisms.

3.1.12 Cohomology and Representations – Character table – Irreducible representations (for groups of moderate order) – KG-module corresponding to an elementary abelian section – p-part of Schur multiplicator, p-cover – Dimensions of ﬁrst and second cohomology groups – Split and non-split extensions of a group by a module (D. Holt’s package) – Schur indices, rewriting representations over minimal ﬁelds

3.1.13 Databases – Transitive groups up to degree 30 (Butler, Hulpke) – Primitive groups up to degree 2499 (Sims, Roney-Dougal & Unger, Roney-Dougal) – Irreducible soluble subgroups of GL(n, p) for pn < 256 (Short) 8 – Almost simple groups of order less than 1.6 × 10 , plus M24, HS, L6(2), J3, McL, Sz(32) stored with their automorphism groups and maximal subgroups – A collection of permutation representations of some sporadic simple groups – Representations of ATLAS groups from the Birmingham ATLAS of ﬁnite group representations (R. Wilson). In V2.17 the database of almost simple groups is augmented by the ability to compute automorphism groups and maximal subgroups for alternating groups up to degree 999, families of low degree classical groups (Holt & Roney-Dougal): Ln(q) for 2 ≤ n ≤ 7 and all q, Ln(2) for n ≤ 14, ± ± S4(q), U3(q) and U4(q) for all q, and the following groups: U6(2), S8(2), S10(2), O8 (2), O10(2), − 3 2 0 S6(3), O7(3), O8 (3), G2(4), G2(5), D4(2), F4(2) , Co2, Co3, He, F i22, Suzi, Ru. 8 Magma Overview V2.18 (December 16, 2011)

3.2 Matrix Groups

Matrix groups may be defined over any ring over which the echelonization of matrices is possible. For example, matrix groups may be defined over function fields K(x). The matrix group facilities are mainly restricted to finite groups since there are, as yet, few algorithms of general interest known for infinite groups. Techniques for working with finite matrix groups divide into methods for groups of small degree and methods for groups of large degree.

3.2.1 Construction

A matrix group is always constructed as a subgroup of the appropriate general linear group, GL(n, R).

– Generators for linear groups: GL(n, R), SL(n, R) where R is a finite field, R = Z, or R = Z/nZ. – Generators for symplectic groups: CSp(n, q), Sp(n, q) – Generators for unitary groups: CU(n, q), GU(n, q), U(n, q) – Generators for orthogonal groups: GO(2n+1, q), SO(2n+1, q), Ω(2n+1, q), GO+(2n, q), SO+(2n, q), Ω+(2n, q), GO−(2n, q), SO−(2n, q), Ω−(2n, q) – Generators for all exceptional families of groups of Lie type – Direct product, tensor wreath product, tensor power, exterior square – Construction of semi-linear groups – Group obtained by applying a homomorphism φ : R → S to the matrix coefficients. – Group obtained by restricting the matrix coefficients to a subring of R.

3.2.2 Arithmetic with Elements

– Random generation of elements (Product-replacement algorithm) – Invariants of a matrix: Trace, determinant, minimal and characteristic polynomials, Jordan form, rational canonical form – Order of a matrix (Leedham-Green algorithm for finite fields) – Test whether a matrix over a number field has infinite order

3.2.3 Actions

– Tests for irreducibility, absolute irreducibility, semi-linearity – Test whether a group over a field of characteristic zero has infinite order – Orbit, stabilizer of a vector or subspace – Orbit representatives and orbit lengths for the action of a matrix group (defined over a prime field) on the k-dimensional subspaces of the natural vector space – Estimate the size of the orbit of a given subspace – Approximation to the stabilizer of a given subspace – Enumerate number of k-dimensional subspaces fixed under the action of an element of GL(d, q) – Homomorphism induced by action of a reducible group on a G-invariant submodule and its quotient module – Homomorphism induced by action on an orbit of vectors or subspaces Magma Overview V2.18 (December 16, 2011) 9

3.2.4 Base and Strong Generating Set For matrix groups of small degree, we use an analogue of the methods used for permutation groups. We try to ﬁnd some sequences of objects (subspaces and vectors) in the underlying vector space that deﬁnes a stabilizer chain which has the property that the basic orbits are not excessively large. Thus, we have a concept of a base and strong generating set (BSGS) similar to that employed in the case of permutation groups. Once such a BSGS is available, analogues of the permutation group backtrack searches for centralizer, normalizer, etc. may be described.

– Random Schreier algorithm for constructing a BSGS – Todd-Coxeter Schreier algorithm for constructing a BSGS – Murray-O’Brien base selection strategy

3.2.5 Elementary Properties With the exception of the ﬁrst, all of the functions listed in this section require the group to be ﬁnite and a BSGS to be computed.

– Whether finite or infinite (only for groups defined over Z or a number field) – Order – Exponent – Whether abelian, nilpotent, soluble, etc. – Whether perfect, simple, etc.

3.2.6 Conjugacy Classes – Testing a pair of elements for conjugacy and ﬁnding a conjugating element – Conjugacy classes of elements (lifting algorithm, classic algorithm) – Power map – Class map, i.e. return the number of the class to which a given element belongs – Class matrix

3.2.7 Subgroup Constructions – Construction of a subgroup in terms of generators – Normal closure, core of a subgroup – Centralizer – Intersection of subgroups – Sylow p-subgroup (reduction algorithm) – Homomorphism induced by action on the cosets of a subgroup – Computing subgroup normalizers

3.2.8 Normal Structure – Derived subgroup – Soluble residual – Centre, Fitting subgroup – Derived series, upper central series, lower central series – Soluble radical, elementary abelian series, p-central series – Composition series, composition factors, chief series – Agemo, omega subgroups (of a p-group) – Jennings series (of a p-group) 10 Magma Overview V2.18 (December 16, 2011)

3.2.9 Standard Quotients – Maximal abelian quotient, elementary abelian quotient – Maximal p-quotient, nilpotent quotient – Maximal soluble quotient – Presentation on strong generators – Quotient by soluble radical

3.2.10 Automorphisms Holt’s algorithm for computing automorphism groups and testing isomorphism may be applied to matrix groups with BSGS.

3.2.11 Cohomology and Representations The Magma machinery for matrix groups together with fast Gr¨obnerbasis techniques (see below) provide a very eﬃcient algorithm for computing a Cohen-Macaulay basis for the ring of invariants together with its syzygies. – Character table – KG-module corresponding to an elementary abelian section – Molien series – Ring of invariants

3.2.12 Aschbacher Analysis

The basic facilities provided by Magma for computing with matrix groups over finite fields depend upon being able to construct a chain of stabilizers. However, there are many examples of groups of moderately small degree where we cannot find a suitable chain. An on-going international research project seeks to develop algorithms to explore the structure of such groups. The main theoretical underpinning of the project comes from the classification by Aschbacher (1984) of the (maximal) subgroups of GL(d, q) into nine families. Much of the research effort to date has been devoted to designing algorithms to decide whether G belongs to one of the eight families whose members have a normal subgroup preserving a “natural linear structure” (the geometric maximal subgroups); here, we plan to exploit this information to explore G further, ultimately producing a composition series for G. – Determine whether a group preserves a form modulo scalars. – The Niemeyer-Prager classical group recognition algorithm as implemented in Magma by Alice Niemeyer and Anthony Pye. – Determine whether a subgroup G of GL(d, q) acts imprimitively on the underlying vector space. a block system, respectively. – Test whether a matrix group G acts as a semilinear group of automorphisms on some vector space. – Test whether a matrix group G preserves a non-trivial tensor product decomposition. – Test whether a matrix group G is tensor-induced. – Search for decompositions (corresponding to certain Aschbacher families) with respect to the normal closure of a supplied subgroup. – The Glasby-Howlett algorithm to decide if the absolutely irreducible group G ≤ GL(d, K) has an equivalent representation over a subfield of K. – Given a group G of d × d matrices over a finite field E having degree e and a subfield F of E having degree f, write G as a group generated by the matrices of G written as de/f × de/f matrices over F . – Roney-Dougal’s algorithm for determining conjugacy of subgroups of GL(d, q) Magma Overview V2.18 (December 16, 2011) 11

In addition, V2.17 of Magma has rotuines to compute all maximal subgroups of Magma’s standard copies of the classical groups of dimension < 12 over ﬁnite ﬁelds. In dimension ≥ 12 this is restricted to computing the geometric maximal subgroups.

3.2.13 Databases of Matrix Groups

– Maximal ﬁnite subgroups of GL(n, Q) for n up to 31

– The finite absolutely irreducible subgroups of GLn(D) where D is a definite quaternion algebra whose centre has degree d over Q and nd ≤ 10 – Irreducible subgroups of GL(n, p) where p is prime and pn < 2500. – Representations of ATLAS groups from the Birmingham ATLAS of finite group representations (R. Wilson).

3.3 Constructive Recognition Given generators for a ﬁnite group, we may be able to determine an isomorphism to a “well- known” group using black-box group methods. Magma is developing its capabilities in this area. The following are available.

– Alternating and Symmetric groups using the algorithm of Bratus & Pak (2000), implemented by Holt. – Alternating and Symmetric groups using the algorithm of Beals et al (2003), implemented by Roney- Dougal. – SL(d, q) and PSL(d, q) groups using the algorithm of Kantor & Seress (2001), implemented by Brooksbank. – Sp(d, q), U(3, q), and U(4, q) algorithms supplied by P. Brooksbank.

3.4 Finitely Presented Groups Given a finitely presented group (fp-group) about which nothing is known, the immediate problems are to determine whether it is trivial, finite, infinite, free, etc. and to determine its finite homomorphic images, finite index subgroups and so on. The central strategy for analyzing an fp-group is to attempt to construct non-trivial homomorphisms, which may be onto an abelian group, p-group, nilpotent group, soluble group, permutation group (the Todd-Coxeter algorithm) or matrix group (vector enumeration).

3.4.1 Free Groups

– Construction – Reduction of a word to normal form – Product, exponentiation, inverse, equality

3.4.2 Construction

– Construction as a quotient of a free group

– Standard groups: Sn, An, dihedral groups, Coxeter groups, braid groups – Permutation groups, matrix groups, polycyclic groups as fp-groups – Direct product, free product – Maximal central extension 12 Magma Overview V2.18 (December 16, 2011)

3.4.3 Arithmetic on Elements – Arithmetic (free reduction only on words) – Substring operations on words – Deﬁnition of and calculation with homomorphisms

3.4.4 Basic Properties Determining global properties of an fp-group is known to be intrinsically diﬃcult. If it is suspected that a given fp-group is ﬁnite, a function is provided that will attempt to determine the order of the group. While this function basically employs coset enumeration, it does so in a fairly sophisticated manner so that it is able to handle groups that are much too large for a coset enumeration of the trivial subgroup to succeed.

3.4.5 Quotients – Abelian quotient, elementary abelian quotient – p-quotient – Process version of p-quotient allowing the user complete control over its execution – Nilpotent quotient (W. Nickel’s algorithm) – Soluble quotient (Plesken-Br¨uckner algorithm) – Process version of the soluble quotient allowing the user complete control over its execution – Natural homomorphism onto any of the above standard quotients – Equivalence classes of homomorphisms to an arbitrary permutation group with application to perfect quotients – Process version of search for equivalence classes of homomorphisms allowing the user complete control over its execution – Kernel of the natural homomorphism onto any standard quotient (provided that the quotient is not too large)

The p-quotient program has been developed over a number of years by George Havas, Mike New- man and Eamonn O’Brien. It has been used to construct p-quotients of composition length several thousand for small primes p. Soluble quotients are computed using Herbert Br¨uckner’s implementation of the Plesken algorithm and is capable of constructing soluble quotients having order in excess of a million. Unlike previous algorithms, no information is required other than the fp-group. The computation of equivalence classes of homomorphisms to a permutation group uses a well known backtrack algorithm. Volker Gebhardt’s implementation of this algorithm is capable of determining all classes of homomorphisms from a 2- or 3-generator group to a permutation group of order up to 108 in reasonable time.

3.4.6 Constructing a Subgroup – Construction of a subgroup in terms of generators – Construction of a subgroup in terms of a coset table – Coset enumeration (Todd-Coxeter procedure) – Process version of coset enumeration allowing the user complete control over its execution – Schreier generators for a subgroup – Presentation for a subgroup (Reidemeister-Schreier rewriting) Coset enumeration is performed using George Havas’s ACE version of the Todd-Coxeter procedure. It has the capability of enumerating up to one hundred million cosets on a suﬃciently large machine. Magma Overview V2.18 (December 16, 2011) 13

3.4.7 Operations on Subgroups of Finite Index The fp-group package also includes a collection of functions for computing with subgroups of (small) ﬁnite index represented by coset tables. Hence the operations in this group assume that the subgroup has ﬁnite index and that it is possible to enumerate its cosets. – Normal closure – Membership and equality of subgroups – Core, intersection and normalizer – All maximal (minimal) overgroups of a subgroup – Test for conjugacy, maximality, normality – Schreier system and Schreier coset map Magma uses Havas’ rewriting Todd-Coxeter to write elements of a subgroup as words in the subgroup generators.

3.4.8 Enumeration of Subgroups

Subgroups of small index may be enumerated using the so-called low index subgroups algorithm. The low index algorithm used in Magma is the backtrack method described by Sims in his book Computation in Finitely Presented Groups, CUP, 1993. – Enumeration of low index subgroups (Sims backtrack algorithm) – Process version of low index subgroups to return subgroups one at a time – Enumeration of low index normal subgroups (Holt’s homomorphism algorithm)

3.4.9 Simplifying a Presentation – Automatic simplification of a presentation – Interactive simplification of a presentation via a Tietze process – Tietze process: Eliminate specified generators – Tietze process: Control of substring searching – Bijection between original and simplified presentations

3.4.10 Actions – Actions on coset spaces (Todd-Coxeter procedure) – Actions on vector spaces (Linton vector enumeration) – Permutation representation on the cosets of a subgroup

3.4.11 Representation Theory It is frequently useful to construct the G-modules corresponding to the conjugation action of a finitely presented group G on an elementary abelian section. This can be useful when attempting to construct normal subgroups of G. – Determine the finite primes dividing the order of the abelian quotient of a subgroup A of a fp-group G – Given subgroups A and B defining an abelian section of G, construct the G-module corresponding to the conjugation action of G on the maximal p-elementary abelian quotient of A/B – Given a map f from a normal subgroup A of G onto the G-module M corresponding to the conjugation action of G on the maximal p-elementary abelian quotient of an abelian section A/B of G and a submodule N of M, compute the preimage of N under f using a fast pullback method.

3.4.12 Databases of Finitely Presented Groups – Fundamental groups of small-volume closed hyperbolic 3-manifolds (Dunﬁeld & Thurston, based on Hodgson & Weeks’ census of manifolds). 14 Magma Overview V2.18 (December 16, 2011)

3.5 Generic Abelian Groups A generic abelian group is a set whose elements form an abelian group with respect to a given law of composition. The user speciﬁes the set A together with functions for composing two elements of A, constructing the inverse of an element of A, and recognizing the identity element of A.

– Deﬁnition as a set with given operations – Arithmetic – Random elements – Order of an element: Baby-step giant-step algorithm; Pollard-rho algorithm – Discrete logarithm: Baby-step giant-step algorithm; Pohlig-Hellman algorithm; Pollard-rho algorithm – Order of the group – Generating set and presentation – Torsion invariants – Construction of subgroups from generators – Sylow p-subgroup – Homomorphisms and isomorphisms

The three major calculations supported are: ﬁnd the order of an element, compute the discrete logarithm of an element relative to a given base and determine the structure of the group. The algorithms used are improvements of those described in J. Buchmann, M.J. Jacobson and E. Teske [10].

3.6 Finitely-Presented Abelian Groups Abelian groups are of interest not only for their intrinsic interest but also because many of the important groups arising in number theory and topology are abelian.

– Construction as a quotient of a free abelian group – Direct product, free product – Arithmetic – Construction of subgroups, quotient groups and complements – Elementary divisors, primary invariants – Factor basis, divisor basis, primary basis – Torsion subgroup, torsion-free subgroup, p-primary component – Homomorphisms: Image, kernel, cokernel – Composition series, maximal subgroups, subgroup lattice (of a finite group) – Character table of a finite group – The group of homomorphisms Hom(A, B), where A and B are finite abelian groups – Abelian quotient of any group (with its natural homomorphism) – Conversion between Z-modules and abelian groups – Functors from rings and fields onto abelian groups The Hermite and Smith normal form algorithms are used to construct a normal form for subgroups and quotient groups of abelian groups. Magma Overview V2.18 (December 16, 2011) 15

3.7 Polycyclic Groups The category described here comprises the family of all groups defined by a polycyclic presentation. Note that such a group may be infinite. Even so, algorithms for element arithmetic analogous to those used for finite soluble groups are available and a growing number of structural computations are possible within such a group.

3.7.1 Polycyclic Groups: Construction and Arithmetic – Construction as a quotient of a free group – Standard groups as polycyclic groups – Permutation groups, matrix groups, abelian groups and ﬁnite soluble groups as polycyclic groups – Nilpotent quotient of a ﬁnitely presented group (W. Nickel’s algorithm) – Direct products – Wreath products – Product, inverse, conjugate, commutator for elements; improved collector, much faster, better com- plexity – Element normal form and equality testing – Element order – Random element generation

3.7.2 Polycyclic Groups: Basic Invariants – Test ﬁniteness, group order – Test for abelian, elementary abelian, cyclic, nilpotent – Nilpotency class – Hirsch number

3.7.3 Polycyclic Groups: Subgroup Constructions – Subgroup and quotient group construction – Normal closure of a subgroup – Conjugation of subgroups – Commutator subgroups – Test subgroup membership and inclusion – Test for normal and central subgroups – Centraliser of elements (in a nilpotent group) – Normaliser and centraliser of subgroups (nilpotent group) – Test for conjugacy of elements and subgroups (nilpotent group) – Intersection of subgroups (nilpotent group)

3.7.4 Polycyclic Groups: Normal Structure – Normal series with free- or elementary-abelian factors – Centre, upper central series – Lower central series, derived subgroup and series – Fitting subgroup, Fitting series – G-module construction from action on free- or elementary abelian sections – Abelian quotient structure 16 Magma Overview V2.18 (December 16, 2011)

3.8 Finite Soluble Groups A large number of efficient algorithms have been developed for computing information about finite soluble groups defined by a polycyclic presentation. The category described here comprises the family of all finite soluble groups defined by polycyclic presentations. Note that while p-groups are not considered as a separate formal category, in many cases more efficient algorithms are employed. Further, some important operations particular to p-groups are described in a separate section.

3.8.1 Construction – Construction of polycyclic presentation for the maximal finite p-quotient of an fp-group (O’Brien’s program) – Construction of polycyclic presentation for the maximal finite soluble quotient of an fp-group (Plesken-Brückner algorithm) – Construction of a polycyclic presentation for a soluble group given as a permutation group or a matrix group – Split and non-split extensions, wreath products – Representation of a soluble group in terms of a SAG-presentation – The soluble groups contained in the Small Groups Library developed by Besche, Eick and O’Brien. This database contains all groups of order up to 2000, except the groups of order 1024, and a number of infinite series of larger groups. In 1991, Leedham-Green with Brownie and Cannon developed an echelonization algorithm capable of constructing a form of polycyclic presentation for a finite soluble group which exhibits much of the structure. This polycyclic presentation (known as a SAG-presentation) is given in terms of generators defining a chain of subgroups which refines a nilpotent series for the group. Each nilpotent section exhibits a lower p-central series for each prime p involved. Finally, the quotient of the group by a term of the nilpotent series splits over the layer below.

3.8.2 Conjugacy Classes – Testing a pair of elements for conjugacy – Conjugacy classes of elements – Power map – Class map, i.e. return the number of the class to which a given element belongs – Class matrix – Exponent

3.8.3 Subgroup Constructions – Construction of subgroups, quotient groups – Normal closure, core of a subgroup – Normalizer, centralizer – Testing subgroups for conjugacy – Intersection of subgroups – Permutation representation on the cosets of a subgroup – System of double coset representatives for a pair of subgroups (Slattery algorithm) – Sylow p-subgroup – Hall π-subgroups, Sylow basis, complement basis – System normalizer, relative system normalizer Magma Overview V2.18 (December 16, 2011) 17

Simple variations of the SAG-algorithm may be used to compute normal closures, the lower central series and the derived series. Sylow p-subgroups, Hall π-subgroups, a Sylow basis, and a complement basis may be read directly from the presentation. The availability of such a presentation together with sophisticated module theory machinery allowed us to design fast algorithms for ﬁnding the centre and maximal subgroups.

3.8.4 Normal Structure – Centre, hypercentre, derived subgroup – Derived series, upper central series, lower central series – Chief series, composition series – Elementary abelian series, p-central series – Fitting subgroup – Frattini subgroup – Normal subgroups

3.8.5 Subgroup Structure – Maximal subgroups – Conjugacy classes of complements of a normal subgroup – Conjugacy classes of subgroups, poset of subgroup classes – Conjugacy classes of subgroups satisfying a condition: Cyclic, elementary abelian, abelian, nilpotent The availability of an SAG-presentation combined with sophisticated module theory machinery allowed us to design fast a algorithm for ﬁnding the maximal subgroups of a soluble group.

3.8.6 Automorphisms and Representations – Automorphism group of a soluble group (M Smith’s algorithm) – Character table (Dixon-Schneider algorithm) – Character degrees (Conlon’s Algorithm) – Modular irreducible representations (Glasby-Howlett algorithm) – Ordinary irreducible representations (Br¨uckner algorithm) – KG-module corresponding to an elementary abelian section

3.9 Finite p-Groups Following the development, in the early 1970’s, of the p-quotient algorithm for constructing polycyclic presentations of a finitely presented p-group, Leedham-Green and others developed an extensive family of elegant and efficient algorithms for finite p-groups. The facilities described here apply to the family of all finite p-groups defined by so-called power-conjugate presentations. Note that as p-groups form a subcategory of the category of finite soluble groups discussed above, most of the soluble group operations apply to p-groups. However, in some cases more efficient algorithms are employed for p-groups than for soluble groups.

3.9.1 Construction – As for ﬁnite soluble groups – Construction of polycyclic presentation for the maximal ﬁnite p-quotient of an fp-group (O’Brien algorithm) – p-group generation (Eamonn O’Brien algorithm) 18 Magma Overview V2.18 (December 16, 2011)

3.9.2 Normal Structure

– As for ﬁnite soluble groups – Agemo, omega subgroups – Jennings series of a p-group

3.9.3 Isomorphisms and Automorphism Groups

– Construct a standard presentation for a p-group – Test two p-groups for isomorphism – Automorphism group of a p-group (E O’Brien’s algorithm)

3.9.4 Character Theory

– Degrees of irreducible characters (Slattery’s algorithm) – Character Table (Conlon’s algorithm)

3.10 Groups Defined by Rewrite Systems This is a category of finitely presented groups where the relations are interpreted as rewrite rules. If the group is defined by a confluent system of rewrite rules then we have a normal form for its elements and hence a solution to the word problem. A group belonging to this category is typically constructed by applying the Knuth-Bendix procedure. As in the case of monoids, Magma uses the Knuth-Bendix developed by Derek Holt as part of his package kbmag.

– Construction of an RWS group from an fp-group using the Knuth-Bendix procedure. Orderings supported include: RT-recursive, recursive, ShortLex, WT-ShortLex and Wreath – Test a rewrite system for confluence – Reduction of a word to normal form – Operations on words: Product, exponentiation, inverse, equality – Enumeration of elements – Test for a group being finite – Definition of homomorphisms whose domain or codomain is an RWS group

3.11 Automatic Groups This category corresponds to short-lex automatic groups. A group is represented by four automata: ﬁrst and second word-diﬀerence machines, a word-acceptor, and a multiplier. These automata are constructed using the Knuth-Bendix procedure. This category is implemented by Derek Holt’s package kbmag.

– Construction of an automatic group from an fp-group using the Knuth-Bendix procedure. – Reduction of a word to normal form – Product, exponentiation, inverse, equality of elements – Enumeration of words without repetition – Test for a group being ﬁnite – Growth function for a group – Deﬁnition of homomorphisms whose domain or codomain is an automatic group Magma Overview V2.18 (December 16, 2011) 19

3.12 Groups with Elements given as Straight-Line Programs This is a class of finitely generated free groups whose elements are represented as “straight-line” programs and which are referred to as SLP-groups for brevity. Typically a SLP-group is used when it is necessary to evaluate long words in a permutation or matrix group G. If G is defined on d generators then a d-generator SLP-group F is defined together with the homomorphism of F onto G which sends the i-th generator of F to the i-th generator of G. Words corresponding to elements of G are built as elements of F where they are represented as expression trees thereby allowing very fast evaluation of long words in G.

– Construction – Arithmetic with straight-line programs – Homomorphism from a blackbox group onto an arbitrary group – Random generation of elements (Leedham-Green & Murray (2002))

3.13 Braid Groups This category comprises the family of all braid groups. Note that this special class of ﬁnitely presented groups has a solvable word problem. Recently, braid groups have received some interest as possible sources of cryptosystems. The Magma implementation by Volker Gebhardt supports both Artin’s original presentation and the band generator presentation introduced by Birman, Ko and Lee. Elements can be deﬁned as words in the generators or as products of simple elements for either presentation. All possible representations of elements can be used simultaneously; conversions are done automatically if necessary, completely transparent to the user.

3.13.1 Constructing and Accessing Braid Groups

– Deﬁnition of a braid group on n strings. – Controlling default presentation, print format for elements and element representation used for group operations.

3.13.2 Constructing and Accessing Elements

– Identity element and fundamental element. – Artin generators and band generators. – Generation of pseudo random elements. – Representations of an element as words and as products of simple elements for Artin presentation and band generator presentation. – Inﬁmum, supremum and canonical length of an element with respect to either presentation.

3.13.3 Normal Forms of Elements

– Left and right normal form with respect to either presentation. – Left and right mixed canonical form with respect to either presentation.

3.13.4 Arithmetic Operations with Elements

– Product, left and right quotient, left and right conjugate of two elements. – Inverse of an element. – Cycling and decycling operation for an element with respect to either presentation. 20 Magma Overview V2.18 (December 16, 2011)

3.13.5 Boolean Predicates – Functions determining whether an element is the identity, simple, or a representative of its super summit class (with respect to either presentation), respectively. – Tests for equality and conjugacy of elements. – Partial orderings of elements with respect to either presentation.

3.13.6 Lattice Operations – GCD and LCM with respect to either presentation and either partial ordering.

3.13.7 Conjugates – Inﬁmum, supremum and canonical length of the super summit class of an element with respect to either presentation. – Computing a representative of the super summit set of an element with respect to either presentation. – Computing the set of positive conjugates of an element with respect to either presentation. – Computing the super summit set of an element with respect to either presentation. – Process versions of the above algorithms for computing positive conjugates and super summit elements.

3.13.8 Homomorphisms – Natural symmetric representation. – Integral and modular Burau representations. – Construction and evaluation of a homomorphism whose domain or codomain is a braid group. Magma Overview V2.18 (December 16, 2011) 21

4 Semigroups and Monoids 4.1 Finitely Presented Semigroups – Construction of fp-semigroups and monoids – Direct product, free product – Arithmetic (free reduction on words only) – Deﬁnition of ideals and subsemigroups – Tietze transformations

4.2 Monoids Defined by Rewrite Systems This is a category of finitely presented monoids where the relations are interpreted as rewrite rules. The most important case is that in which the monoid is defined by a confluent system of rewrite rules. A monoid of this category is typically constructed by applying the Knuth-Bendix procedure to a finitely presented monoid. Magma uses the Knuth-Bendix developed by Derek Holt in his package kbmag. – Construction of an RWS monoid from an fp-monoid using the Knuth-Bendix procedure. Orderings supported include: RT-recursive, recursive, ShortLex, WT-ShortLex and Wreath – Test a rewrite system for confluence – Reduction of a word to normal form – Operations on words: Product, exponentiation, equality – Test for a monoid being finite – Enumeration of elements – Definition of homomorphisms whose domain or codomain is an RWS monoid 22 Magma Overview V2.18 (December 16, 2011)

5 Rings and their Fields

This section is concerned with ﬁelds (mainly local and global arithmetic ﬁelds), their rings of integers and valuation rings.

• The rational field Q and its ring of integers Z • Residue class rings of Z • Univariate polynomial rings • Finite fields • Number fields and their orders • Rational function fields • Algebraic function fields • Valuation rings • Real and complex fields • p-adic Rings and their extensions • General local fields • Power series rings and Laurent series rings

In the case of arithmetic fields, the major facilities include: – Construction of a basis for the maximal order(s) – Decomposition of ideals into prime ideals – Recognition of principal ideals – Class group, divisor class group and ray class groups – Fundamental units – Constructive class field theory – Galois theory, subfields and automorphisms – Places and completions – Cohomology of number fields and their completions Magma Overview V2.18 (December 16, 2011) 23

5.1 The Rational Field and its Ring of Integers

5.1.1 Arithmetic – Multiple precision integer arithmetic – Integer multiplication via classical, Karatsuba, Toom and Schönhage-StrassenFFT methods – Integer division via classical, Karatsuba, Toom and Schönhage-StrassenFFT methods – Greatest common divisor via Weber Accelerated GCD and Schönhagealgorithms – Extended Greatest common divisor via Lehmer and Schönhagealgorithms – Alternative representation of integers in factored form – Arithmetic functions: Jacobi symbol, Euler φ function, etc. Magma uses portions of the GMP package for the base classical, Karatsuba and Toom and Schönhage-StrassenFFT-based algorithms (for which GMP is the state of the art). Magma also contains an asymptotically-fast integer (and polynomial) division algorithm which reduces division to multiplication with a constant scale factor that is in the practical range. Thus division of integers and polynomials are based on the fast multiplication methods when applicable. Finally, Magma contains implementations of the fast classical Lehmer extended GCD (‘XGCD’) algorithm (which is about 5 times faster than the Euclidean XGCD algorithm) and the Schönhage recursive (“half-GCD”) algorithm, yielding asymptotically-fast GCD and XGCD algorithms.

5.1.2 Residue Class Rings of Z A quotient ring Z/hmi of Z is trivially represented by the integers taken modulo m. In the Magma implementation, m may be taken to be a long integer. – Arithmetic – Testing elements for: nilpotency, primitivity, regularity, zero-divisor – Order of a unit – Gcd and lcm – Location of a primitive element – Unit group – Functor from additive group to an object in the category of abelian groups – One or all square roots of an element – Linear algebra

5.1.3 Primality and Factorization – Probabilistic primality testing (Miller-Rabin) – Rigorous primality testing (Morain’s Elliptic Curve Primality Prover) – Primality certificates; Verification of certificates – Generation of primes – Elementary factorization techniques: Trial division, SQUFOF, Pollard ρ, Pollard p − 1 – Elliptic curve method for integer factorization (A. Lenstra) – Multiple prime multiple polynomial quadratic sieve algorithm for integer factorization (A. Lenstra) – Database of factorizations of integers of the form pn ± 1

The Elliptic Curve Primality Prover (ECPP) designed and implemented by Fran¸coisMorain at INRIA is installed in Magma. This provides fast rigorous primality proofs for integers having several hundred digits. The primality of a 100 digit integer is established in 24 seconds (on a Sun 200Mhz SPARC workstation 2). Paul Zimmermann’s eﬃcient GMP-ECM package is included for integer factorization. The GMP-ECM package also provides fast p − 1 and p + 1 integer factorisation algorithms. 24 Magma Overview V2.18 (December 16, 2011)

5.1.4 The Number Field Sieve

Magma provides an experimental implementation of the fastest general purpose factoring algorithm known: the number field sieve (NFS). The implementation can be used for both general number field and special number field factorizations: the only difference is in the polynomial selection. Presently, Magma does not provide a function to choose a good polynomial for a particular number to be factored. However, Magma does provide some functions that are useful for the implementation of the polynomial selection algorithms developed by Peter Montgomery and Brian Murphy.

5.2 Dirichlet Characters The Dirichlet characters package provides support for computing with the group of homomorphism (Z/NZ)∗ → R∗, where R is a ring. New functionality also exists for Dirichlet (and Hecke) characters over number ﬁelds. ∗ – Dirichlet group over a ﬁeld K as the set of rational characters from (Z/NZ)∗ to K . – Computation of group generators, enumeration of elements, and representatives of Galois-conjugacy classes of characters. – Construction and evaluation of Dirichlet characters. – Invariants of characters, such as their conductor, modulus, and order. Magma Overview V2.18 (December 16, 2011) 25

5.3 Univariate Polynomial Rings A polynomial ring may be formed over any ring, including a polynomial ring. Since computational methods for univariate polynomial rings are often much simpler and more efficient than those for multivariate rings (especially over a field), we discuss the two cases separately. In this section, the symbol K, appearing as a coefficient ring, will denote a field.

5.3.1 Creation and Ring Operations

– Creation of a polynomial ring – Deﬁnition of a ring map – Kernel of a ring map – Determining whether a ring map is surjective – Determining whether a ring map is an isomorphism

5.3.2 Creation of Special Polynomials

– Orthogonal polynomials: Bernoulli polynomial – Orthogonal polynomials: Chebyshev polynomials of the ﬁrst and second kinds – Orthogonal polynomials: Chebyshev polynomials of types T and U – Orthogonal polynomials: Gegenbauer polynomial, Hermite polynomial – Orthogonal polynomials: Generalised Laguerre polynomial – Orthogonal polynomials: Legendre polynomial – Binomial polynomial – Conway polynomial of degree n over GF (p) – Primitive polynomial of degree n over GF (q) – Cyclotomic polynomial of order n – Permutation polynomials: Dickson polynomials of the ﬁrst and second kinds

5.3.3 Arithmetic with Polynomials

– Polynomial product via classical, Karatsuba and Schönhage-StrassenFFT algorithms – Polynomial quotient via classical and Karatsuba algorithms – Pseudo quotient and remainder – Modular exponentiation and inverse – Modular composition over GF (q) (Brent-Kung) – Norms – Differentiation and integration – Evaluation and interpolation – Properties: prime, primitive, separable, permutation – Properties: a unit, a zero-divisor, nilpotent – Properties: Galois groups of square-free polynomials over the integers, number fields or prime characteristic function fields – For polynomials over Z: splitting field and solvability by radicals 26 Magma Overview V2.18 (December 16, 2011)

Magma employs asymptotically fast algorithms for performing arithmetic with univariate polynomials over certain rings. These include two FFT-based methods for multiplication: the Schönhage- Strassen FFT method for situations where the coefficients are large compared with the degree, and the small-prime modular FFT with Chinese remaindering method for where the coefficients are small compared with the degree. These methods are applied to multiplication of polynomials over Z, Q, Z/mZ and GF (q). For some coefficient rings, at least one of the FFT methods outper- forms the Karatsuba method for polynomials having degree as small as 32 or 64; for each of the coefficient rings listed, the FFT method beats the Karatsuba method for degree 128 or greater. An asymptotically-fast division algorithm (which reduces division to multiplication) is also used for polynomials over all coefficient rings. Magma Overview V2.18 (December 16, 2011) 27

5.3.4 GCD and Factorization

– Resultant, discriminant (sub-resultant algorithm, Euclidean algorithm) – Greatest common divisor, extended greatest common divisor – Extended greatest common divisor – Hensel lift – Squarefree factorization – Distinct degree factorization – Factorization over GF(q): Small ﬁeld Berlekamp, large ﬁeld Berlekamp, Shoup algorithms – Factorization over Z and Q: van Hoeij algorithm

– Factorization over Qp and all local rings and ﬁelds – Factorization over Q(α): Trager algorithm – Factorization over any ﬁnitely generated extension of GF(p) and Q – Galois group computation for arbitrary polynomials in Q[t]

Greatest common divisors for polynomials over Z are computed using either a modular algorithm or the GCD-HEU method, while for polynomials over a number field, the modular method is used. For polynomials over an algebraic function field, an evaluation/interpolation algorithm of Allan Steel is used. Factorization of polynomials over Z uses the exciting new algorithm of Mark van Hoeij, which efficiently finds the correct combinations of modular factors by solving a Knapsack problem via the LLL lattice-basis reduction algorithm.

5.3.5 Arithmetic with Ideals

– Construction of ideals and subrings (over K) – Construction of quotient rings – Arithmetic with ideals (over K) – Properties of an ideal: Maximal, prime, primary

5.4 Residue Class Rings of Univariate Polynomial Rings

– Arithmetic with elements – Construction of ideals and subrings (over K) – Construction of quotient rings – Arithmetic with ideals (over K) 28 Magma Overview V2.18 (December 16, 2011)

5.5 Finite Fields 5.5.1 Construction

– Construction of fields GF(p), p large; GF(pn), p small and n large – Optimized representations of GF(pn) in the case of small p and large n – Database of sparse irreducible polynomials over small finite fields to very high degrees (see details in Subsec. 22.3 on page 118) – Special optimized packed arithmetic for fields of characteristic 2 and 3. – Construction of towers of extensions – Construction of subfields – Compatible embedding of subfields – Enumeration of irreducible polynomials – Fast embedding and isomorphism algorithm of E. Rains. The finite field module uses different representations of finite field elements depending upon the size of the field. Thus, in the case of small to medium sized fields, the Zech logarithm representation is used. For fields of characteristic 2, a packed representation is used (since V2.4), which is very much faster than the representation used in previous versions of Magma. For a large degree extension K of a (small) prime field, K is represented as an extension of an intermediate field F whenever possible. The intermediate field F is chosen to be small enough so that the fast Zech logarithm representation may be used. Thus, Magma supports finite fields ranging from GF(2n), where the degree n may be a ten thousand or more, to fields GF(p), where the characteristic p may be a thousand-bit integer. The finite field code is highly optimized for very small finite fields and especially for linear algebra over such fields. A noteworthy feature of the facility is that no matter how a field is created, its embedding into an overfield may be determined. One may create and work within a lattice of subfields with create ease. The system is described in detail in a paper [6]. Using the fast algorithm of Rains, embedding the field GF(21000) in GF(22000) now takes 0.3 seconds on a 2.4GHz Opteron, compared to about 20 minutes with the old root finding method.

5.5.2 Arithmetic

– Trace and norm; Relative trace and norm – Solving Norm and Hilbert-90 equations – Order of an element – Characteristic and minimum polynomials – Testing elements for: Normality, primitivity – Fast powering via Frobenius map – Construction of primitive and normal elements

5.5.3 Roots and Polynomial Factorization

– Square root: Tonelli-Shanks method for GF(p) – n-th root – One root of a polynomial; All roots root of a polynomial – Polynomial factorization: Small field Berlekamp, large field Berlekamp – Polynomial factorization: Shoup algorithm – Construction of a splitting field – Factorisation over splitting field Factorization of polynomials over GF(q) for large q uses Shoup’s algorithm. On a 64-bit 2.4GHz Opteron, the n = 2048 polynomial from the von zur Gathen challenge benchmarks (of degree 2048 with sparse coefficients modulo a 2050-bit prime) is factored by Magma in 1300 seconds. Magma Overview V2.18 (December 16, 2011) 29

5.5.4 Discrete Logarithms

– Shanks baby-step giant-step algorithm – Pollard rho algorithm – Polhig-Hellman algorithm – Index calculus method using either the Gaussian integer sieve or linear sieve (for prime fields GF(p)). – Index calculus method using Emmanuel Thomé’simplementation of Coppersmith’s method (for fields GF(2k)).

The index calculus method applied to an arbitrary ﬁeld GF(p), where p is a 100-bit prime such that (p − 1)/2 is prime (the worst case), takes 10 seconds to perform the sieving and about 0.8 seconds to compute an individual logarithm. For a 20-decimal-digit prime p such that (p − 1)/2 is prime, Magma takes only 1 second for the sieving and about 0.3 seconds to compute an individual logarithm.

5.5.5 Derived Structures

– Unit group – Additive group – Field as an algebra over a subﬁeld 30 Magma Overview V2.18 (December 16, 2011)

6 Global Arithmetic Fields

6.1 Number Fields and their Orders Number fields in Magma are (abstract) finite extensions of Q or other number fields that can be constructed in a large number of different ways to faciliate a multitude of internal and external applications. For example number fields can be defined by specifying a single irreducible polynomial, as subfields of other number fields, via Galois correspondence, from characters, from a range of geometric objects or simply as cyclotomic fields of a given order. Number fields and their orders support a huge number of algorithms, starting from simple arithmetic, the computation of integral closures (maximal orders) and class and unit group to sophisticated tools for Galois cohomology and Diophantine equations. As well as state of the art algorithms for general numebr fields, Magma also contains special methods for quadratic and cyclotomic fields. Facilities for general number fields have been developed in a joint project with the KANT group in Berlin. Number fields interact easily with their completions thus allowing a modern, local-global approach to a large number of problems.

6.1.1 Number Fields

– Arithmetic of elements – Construction of equation orders, maximal orders, arbitrary orders – Simple and relative extensions, extensions defined by several polynomials – Subfields – Transfer between relative and absolute representations – Discriminant, reduced discriminant, signature – Factorization of polynomials over number fields – Completions of absolute fields – Number fields with arbitrary bases – Computation of Hilbert class fields and general class fields – Representation of a number field as a vector space or algebra over a given coefficient field – Automorphism group, Galois group of the normal closure – Action of the automorphisms on ideals and ideal classes – Dirichlet and Hecke character group as duals of residue rings and ray class groups – Basic Galois-Cohomology, computations in H1 and H2

6.1.2 Orders and Fractional Ideals – Multiple relative extensions – Maximal order, integral basis (Round 2 and Round 4 algorithms for absolute fields, Round 2 for relative extensions and special methods for Kummer extensions) – Suborders, extension orders – Construction of integral and fractional ideals – Ideal arithmetic: product, quotient, gcd, lcm, colon ideal – Determination of whether an ideal is: integral, prime, principal – Decomposition of primes – Valuations of order elements and ideals at prime ideals – Completions at prime ideals – Factorization of an ideal – Residue field of an order modulo a prime ideal – Residue class ring of an order modulo an arbitrary ideal – Completion of absolute maximal orders at finite primes Magma Overview V2.18 (December 16, 2011) 31

6.1.3 Invariants

– Class group: Conditional (GRH) and unconditional algorithms – Unit group: Conditional (GRH) and unconditional algorithms – S-Unit group for arbitrary (ﬁnite) set S of primes – Picard group (ring class group) for non-maximal orders – Regulator – Exceptional units – Ray class groups, unit group of ray class rings of absolute maximal orders. – p-Selmer groups

6.1.4 Diophantine (and other) Equations

– Norm equations, relative norm equations (both in the ﬁeld and the order case, testing for local solubility) – Simultaneous norm equations, splitting of 2-cocycles – Thue equations – Unit equations – Index form equations – Integral points on Mordell curves – Hilbert 90

6.1.5 Quadratic Fields

– All functionality of number fields √ – Euclidean structure of Q( d), for d = −1, −2, −3, −7, −11, 2, 3, 5, 13. – Class number (Shanks’ algorithm) – Ideal class group (Buchmann’s method) – Fundamental unit, conductor – Solution of norm equations (Cornaccia’s algorithm for imaginary fields and Cremona’s conics for real quadratic fields) – Facilities for binary quadratic forms (see Lattices and Quadratic Forms) – 2-class group using the Bosma–Stevenhagen method

6.1.6 Cyclotomic Fields

– All functionality of number fields – Sparse representation for large fields – Conductor and cyclotomic order – Cyclotomic subfields – Creation of roots of unity – Minimization of elements into smaller fields – Conjugation and complex conjugation 32 Magma Overview V2.18 (December 16, 2011)

6.2 Galois Theory of Number Fields Magma contains a rich set of commands to use and analyze the Galois structure of number fields. Starting with Klüners’or Klünersand van Höij’ssubfield algorithm that allows the computation of all subfields of extensions of Q without the knowlegde of the Galois group, a generalization by Fieker and Klünersof Staudhar’s method to compute the Galois group of any rational polynomial (no degree restriction, reducible polynomials as well) to Klünersalgorithms for automorphisms of abelian fields. Magma has been able to compute Galois groups of irreducible polynomials of degree > 50 and of reducible polynomials of degree > 25. – Determination of subfields – Automorphism groups of normal and abelian fields – Action of automorphisms on ideals, class group – Isomorphisms and embeddings of number fields – Galois group of number fields with no degree restriction. – Galois correspondence: fixed fields, fixed groups – Solvability by radicals, virtual work in the splitting field – Ramification theory – Action on S-Units

The computation of the Galois group of the degree 32 polynomial f(f(f(f(x)))) for f := x2 − 2 takes about 5 seconds, to solve (by radicals) a polynomial x6 − 3x5 − 2x4 + 9x3 − x2 − 4x + 1 with Galois group 6T11 of order 48 takes also about 5 seconds.

6.3 Class Field Theory of Number Fields Class field theory is one of the most important results in number theory of the 20th century. While thought to be totally theoretical for a long time, it is now very practical and used in a growing number of applications from tabulating fields to Diophantine equations and representation theory of finite groups. The core of Magma’s class field theory is Fieker’s algorithm to compute defining equations for a class field that is parametrized as a quotient of a ray class group. While analytic contructions are also available (over imaginary quadratic fields), this algebraic method applies to all fields. – Ray class groups modulo any integral ideal – Computation of defining equations of class fields – Test if a class field is normal, central or abelian without computing defining equations – Special integral closure algorithms based on Kummer theory – Analytical construction of Hilbert Class polynomials over imaginary quadratic fields – Computation of Hilbert class fields and ring class fields – The norm group of an abelian number field can be computed – Norm symbols, Artin-map is available – Solvability of norm equations can be tested, extending the Hasse-Norm-Theorem – Extension of automorphisms of the base field – Galois cohomology: action on Ray class groups – Galois cohomology: decide if a 1 or 2-cocycle is trivial, explicit splitting of cocycles – Grunwald-Wang theorem: find a cyclic extension having prescribed degrees at a finite number of places. – Functionality with Hecke Grössencharacters in CM fields Magma Overview V2.18 (December 16, 2011) 33

6.4 General Algebraic Function Fields A general algebraic function field F/k of n variables over a field k is a field extension F of k such that F is a field extension of finite degree of k(x1, . . . , xn) for elements xi ∈ F which are algebraically independent over k.

6.4.1 Rational Function Fields

Given any ﬁeld k and indeterminates x1, . . . , xn, the user may form the ﬁeld of rational functions k(x1, . . . , xn) as the localization of the polynomial ring k[x1, . . . , xn] at the prime ideal hx1, . . . , xni.

– Creation of a rational function ﬁeld of a given rank over a given ring – Retrieval of the ring of integers, coeﬃcient ring and rank – Ring predicates – Arithmetic – Numerator and Denominator – Degree and weighted degree – Evaluation – Derivative – Partial fraction expansion – Partial fraction decomposition (squarefree or full factorization)

6.4.2 Algebraic Function Fields Within Magma, algebraic function fields of one variable can be created by adjoining a root of an irreducible, separable polynomial in k(x)[y] to the rational function field k(x). If k is a finite field, the function field is said to be global. An algebraic function field can be extended to create fields of the form k(x, a1, . . . , ar) where each extension occurs by adjoining a root of an irreducible and separable polynomial. Extensions may be formed using several polynomials simultaneously giving a non simple representation.

– Creation of simple, relative and non simple extensions and mixed towers thereof – Creation of extensions of the constant field using bivariate polynomials – Retrieval of information defining the field – Exact constant field and genus – Change of representation from finite degree extensions to infinite and vice versa – Change of coefficient field to one lower in the extension tower – Computation of subfields and automorphisms – Homomorphisms from function fields into any ring by specifying the image of the primitive element and an optional map on the coefficient field – Computation of Galois Groups of simple extensions of a function field with no degree restriction (and of squarefree separable polynomials over global function fields) – k-Automorphisms, isomorphism testing and embeddings – L-polynomial and ζ-function – Construction of a function field with an extended constant field – Construction of Artin-Schreier-Witt extensions from finite dimensional Witt-vectors – Constructive class field theory using both algebraic and analytic (Drinfeld modules) methods 34 Magma Overview V2.18 (December 16, 2011)

6.4.3 Orders of Algebraic Function Fields – Finite and infinite equation orders – Finite and infinite maximal orders using the Round 2 algorithm for extensions which are not Kummer or Artin–Schreier, a basis for a p-maximal order of a Kummer extension and a maximal order of an Artin–Schreier extension can be written down. – Creation of orders whose basis is a transformation of an existing order – Integral closure – Basis of the order with the option to have the elements returned in a specified ring – Simplification of an order to a transformation of its equation order – (S-)Unit Group and unit rank, independent and fundamental units and regulator – Ideal class group for maximal orders – Ring predicates – Basis size reduction for finite, simple, non relative orders

6.4.4 Elements of Algebraic Function Fields and their Orders Elements of function fields and their orders have 3 different representations. These representations are implemented generally for function field elements and number field elements. Standard elements are represented using coefficients of the basis elements. Elements of orders (and fields) with a power basis are represented using a polynomial representation. Elements of all orders or fields may have a product representation, being thought of as a formal product of a list of elements each to the power of some exponent. This can be a great advantage when the element is prohibitively large when represented using coefficients. – Arithmetic and modular arithmetic – Predicates – Creation of random elements and conversion to and from sequences – Norm and trace with respect to any given coefficient ring – Representation matrix, minimal and characteristic polynomials – Numerators and Denominators with respect to a given order – Module generated by a sequence of elements – Strong approximation theorem – Power series expansion, mapping of elements into completions

6.4.5 Ideals of Orders of Algebraic Function Fields – Creation of ideals from generators or a basis – Arithmetic – Roots of ideals – Predicates for integrality, prime, principal zero and one ideals – Predicate for prime ideals determining the type of ramification – Intersection, GCD and LCM – Factorization – p-radicals and p-maximal orders – Taking valuations of elements and ideals at prime ideals – Denominator – Retrieving basis and generators – Residue class field and the map to and from the order into it – Completions – Ramification and inertia degree Magma Overview V2.18 (December 16, 2011) 35

6.4.6 Places of Algebraic Function Fields – Creation of places as zeros and poles of elements of a field – Creation from prime ideals – Creation of random places of global fields – Creation of places of a given degree of global fields – Decomposition of places – Arithmetic – Residue class field, lifting elements out of and evaluating functions into – Valuation of elements and expanding elements at a place – Completion of fields and orders at places of any degree – Ramification and inertia degree – Retrieval of generators and a uniformizing element – Weierstrass places – Counting the number of places of a given degree over the exact constant field of global fields – The Serre and Ihara bounds on the number of places of degree 1 over the exact constant field of global fields

6.4.7 Divisors of Algebraic Function Fields – Creation from places, elements and ideals – Canonical and different divisor – Arithmetic including GCD and LCM – Support and Degree – Numerator and Denominator – Testing for properties of effective, positive, principal, special and canonical – Riemann–Roch space L(D) of a divisor D, given by a k-basis of algebraic functions – Reduction of a divisor – Index of Speciality – Gap numbers, ramification divisors, Wronskian orders and Weierstrass places – Parametrization of a field at a divisor – Number of smooth divisors of global fields

6.4.8 Differentials of Algebraic Function Fields – Creation of a differential space – Creation of differentials from field elements – Arithmetic – Valuation of a differential at a place – Divisor of a differential – Differential spaces and bases for given divisors – Space and basis of holomorphic differentials of a field – Differentiations of elements of a function field – Residue of a differential at a place of degree one – Cartier operator and representation matrix of the Cartier operator (global case) – Module generated by a sequence of differentials 36 Magma Overview V2.18 (December 16, 2011)

6.4.9 Divisor Class Groups for Global Algebraic Function Fields – Bounds on the generation of the class group – Computation of the class number and approximations to it – Construction of the divisor class group, structure of the divisor class group, representation of divisor classes as abelian group elements – S-class group, S-units and S-regulator for a ﬁnite set of places S – Exact sequence 0 → U(S) → F × → Div(S) → Cl(S) → 0 – Image and preimage computation possible for the maps of the exact sequence – Similar functionality for the ideal class group of the ﬁnite maximal order – p-rank of the divisor class group (separate method) and Hasse–Witt invariant – Tate–Lichtenbaum pairing – Global units

6.4.10 Class Field Theory for Algebraic Function Fields – Ray divisor class groups – Defining equation for class fields – Conductor and norm group – Genus, discriminant, number of places of given degree – Decomposition type of places of the base field – Exact constant field – Drinfeld modules of rank 1, rings of twisted polynomials

The development of this module is a joint project with the KANT group. Magma Overview V2.18 (December 16, 2011) 37

6.5 Algebraically Closed Fields Algebraically closed ﬁelds (ACF’s) have the property that they always contain all the roots of any polynomial deﬁned over them.

– Construction of algebraic closures over a finite field, the rational field or a rational function field of any characteristic – Automatic extension of the field by the roots of any polynomial over the field, and operations on conjugates of roots – Basic arithmetic – All standard algorithms for rings over generic fields work over such fields – Minimal polynomial – Simplification of the field – Construction of the corresponding absolute field together with the isomorphism – Pruning of useless variables and relations

It is not possible to construct explicitly the closure of a field, but the system works by automatically constructing larger and larger algebraic extensions of an original base field as needed during a computation, thus giving the illusion of computing in the algebraic closure of the base field. A similar system was suggested by D. Duval and others (the D5 system [16]), but this has difficulty with the parallelism which occurs when one must compute with several conjugates of a root of a reducible polynomial, leading to situations where a certain expression evaluated at a root is invertible but evaluated at a conjugate of that root is not invertible. The system developed for Magma by Allan Steel avoids these problems, and is described in [41, 42]. Consequently, ACF’s behave in the same way as any other field implemented in Magma; all standard algorithms implemented for generic fields and which use factorization work without change (for example, the Jordan form of a matrix). The system avoids factorization over algebraic number fields when possible, and automatically splits the defining polynomials of a field when factors are found. The field may also be simplified and expressed as an absolute field. Especially significant is also the fact that all the Gröbner basis algorithms work well over ACF’s. One can now compute the variety of any zero-dimensional multivariate polynomial ideal over the algebraic closure of its base field. Puiseux expansions of polynomials are now also computed using an algebraically closed field. Since V2.13, one may construct the algebraic closure over a finite field or a rational function field of any characteristic. For rational function fields of very small characteristic, inseparable field extensions are handled properly (see [42] for details). 38 Magma Overview V2.18 (December 16, 2011)

7 Local Arithmetic Fields Magma Overview V2.18 (December 16, 2011) 39

7.1 Discrete Valuation Rings Valuation rings are available for the rational field and for rational function fields. For rational function fields, given an arbitrary monic irreducible polynomial p(x) ∈ K[x], the valuation ring is

f(x) O = { : f(x), g(x) ∈ K[x], p(x)6 | g(x)}. p(x) g(x)

Valuations corresponding both to an irreducible element and to ∞ are allowed.

– Valuation ring corresponding to the discrete non-Archimedean valuation vp of Q

– Valuation ring corresponding to the discrete non-Archimedean valuation vp of a rational function ﬁeld

– Valuation ring corresponding to the valuation v∞ of a rational function ﬁeld – Arithmetic – Euclidean norm, valuation – Greatest common divisor 40 Magma Overview V2.18 (December 16, 2011)

7.2 The Real and Complex Fields The real and complex ﬁelds are diﬀerent from most structures in that exact computation in them is almost never possible.

– Arithmetic – Square root, arithmetic-geometric mean – Continued fraction expansion of a real number – Constants: π, Euler’s constant, Catalan’s constant – Logarithm, dilogarithm, exponential – Trigonometric functions, hyperbolic functions and their inverses – Bernoulli numbers – Γ function, incomplete Γ function, complementary incomplete Γ function, logarithm of Γ function – J-Bessel function, K-Bessel function – U-conﬂuent hypergeometric function – Logarithmic integral, exponential integral – Error function, complementary error function – Dedekind η function – Jacobi sine theta-function and its k-th derivative Γ0(x) – Log derivative (ψ) function, i.e, Γ(x) – Riemann-ζ function – Polylogarithm, Zagier’s modiﬁcations of the polylogarithm

– Weber’s f-function, Weber’s f2-function, j-invariant – Integer polynomial having a given real or complex number as an approximate root (Hastad, Lagarias and Schnorr LLL-method) – Roots of an exact polynomial to a specified precision (Schönhagesplitting circle method) – Summation of a series (Euler-Wijngaarden method for alternating series) – Numerical integration of a function (Romberg-type methods) The real and complex fields in Magma are based on the GMP, MPFR and MPC packages. Some of the transcendental functions as well as root finding is based on code developed for PARI by Henri Cohen and others.

7.3 Newton Polygons – Construction of a newton polygon: Compact, infinite or including the origin – Construction of newton polygons from different types of data: f ∈ k[x, y], f ∈ k hhxii [y], f ∈ k[y] and some prime object, a finite set of points, a finite set of faces (weighted dual vectors) – Finding faces, vertices and slopes – If polygon is derived from a polynomial f, finding restrictions of f to faces – Locating a given point relative to a newton polygon – Giving the valuations of the roots of a polynomial (with respect to a prime if not implicit) Newton polygons can be used with polynomials over series rings in order to find roots of the polynomial. – Walker’s [48] algorithm for computing Puiseux expansions – Duval’s [17] algorithm for computing Puiseux expansions Magma Overview V2.18 (December 16, 2011) 41

7.4 p-adic Rings and their Extensions A p-adic ring arises as the completion of the ring of integers at a prime while a local field arises as the completion at a prime ideal of a number field. Magma supports both fixed and free precision models, allowing the user to trade an increase in speed for automated precision management.

7.4.1 Construction – Construction of a p-adic ring or field (via polynomials or as completions) – Unramified extension of a local ring or field – Totally ramified extension of a local ring or field – Ring of integers of a local field – Field of fractions of a local ring – Change precision of a ring, field or element – Computation of a splitting field of an integral polynomial over an p-adic ring. – Enumeration of extensions of a given degree.

A local ring is a finite degree extension of a p-adic ring and may be either ramified or unramified or both. Any arbitrary tower of extensions can be constructed, as long as each step is either ramified or unramified or both.

7.4.2 Arithmetic – Arithmetic operations – Valuation of an element – Norm and trace of an element – Logarithm, exponential of an element – Square root, n-th root of an element – Minimal polynomial of an element over the p-adic subring or ﬁeld – Image of an element under a power of the Frobenius automorphism – Linear algebra over local rings and ﬁelds

7.4.3 Polynomial Factorization – Polynomial algebra over local rings and fields – Greatest common divisor of two polynomials – Hensel lifting of the factors of a polynomial – Hensel lifting of the roots of a polynomial – Test a polynomial for irreducibility – Roots of a polynomial over a local ring or field – Factorization of polynomials over local ring or field

7.4.4 Class ﬁeld theory – Unit group and norm group – Deﬁning equations for abelian extensions 42 Magma Overview V2.18 (December 16, 2011)

7.5 General Local Fields In addition to the p-adic rings and their (ramified and unramified) extensions Magma contains local fields which are defined as an extension of another local field by any irreducible polynomial.

– Construction of local fields as an extension by any irreducible polynomial – Calculation of subfields – Isomorphisms to extensions of p-adic fields – Construction of elements and basic arithmetic with those elements – Homomorphisms from local fields – Automorphism group and subgroups and fixed fields of these groups – Computation of an existing field as a extension of another field

7.6 Galois Rings Magma provides facilities for computing with Galois rings. The features are currently very basic, but advanced features will be available in the near future, including support for the creation of subrings and appropriate embeddings, allowing lattices of compatible embeddings, just as for finite fields. Because of the valuation defined on them, Galois rings are Euclidean rings, so they may be used in Magma in any place where general Euclidean rings are valid. This includes many matrix and module functions, and the computation of Gröbnerbases. Linear codes over Galois rings will also be supported in the near future.

Features: – Creation of a default Galois ring (using a default defining polynomial). – Creation of a Galois ring by a specified defining polynomial. – Basic structural operations and arithmetic. – Euclidean operations. Magma Overview V2.18 (December 16, 2011) 43

7.7 Power, Laurent and Puiseux Series Rings Magma contains an extensive package for formal power series. The fact that we may only work with a finite number of terms, n say, of a power series, i.e., a truncated power series, is made precise by noting that we are working in the quotient ring R[[x]]/hxn+1i, for some n, rather than in the full ring R[[x]]. Provided this is kept in mind, calculations with elements of a power series ring (though not field) are always precise. Given a field K, a field of Laurent series K((x)) is regarded as the localization of the power series ring K[[x]] at the ideal h0i. More simply, it is the field of fractions of K[[x]]. Since elements of such a field are infinite series, calculation is necessarily approximate. A power series ring R[[x]] is regarded as the completion of the polynomial ring R[x] at the ideal h0i. Puiseux series with arbitrary fractional exponents are also supported (since V2.4).

– Arithmetic – Inversion of units – Derivative, integral – Square root, valuation – Exponentiation, composition, convolution, reversion – Power series expansions of transcendental functions – R[[x]]/hxn+1i as an algebra over R – Factorization of polynomials over series rings – Unramiﬁed and ramiﬁed extensions of series rings

7.8 Lazy Power Series Rings These power series rings contain only series of infinite precision. All coefficients of such series are computable but only finitely many will be known.

– Creation of rings and elements – Arithmetic of elements – Retrieval of coeﬃcients – Printing some speciﬁed terms of a series – Simple predicates on series – Derivative, integral and evaluation of series 44 Magma Overview V2.18 (December 16, 2011)

8 Linear Algebra and Module Theory

• Matrix operations • Vector spaces • Free modules • Modules over Dedekind domains

8.1 Matrices

In this section we list the basic facilities available for computing with matrices over various rings. These algorithms underpin the vector space and module theory machinery.

8.1.1 Representation of Matrices

– Optimal packed representation for matrices over GF(q), using specialized bit operations, for q = 2, 4, 8, 16. – Special packed representation for matrices over GF(q), using bit operations, for q = 3, 5, 7, 9. – Efficient packed representation for matrices over all other small finite fields, using lookup tables for vector operations. – Fraction-free algorithms for matrices over Q, reducing computations to those over Z. – Internal modular representations of matrices over Z or Q, for several algorithms. – Fraction-free algorithms for matrices over rational function fields, reducing computations to those over polynomial rings. – Input of matrices using the “Cambridge” compact format

8.1.2 Arithmetic

– Multiplication: fast baby-step/giant-step algorithm for small characteristic finite fields – Multiplication: Strassen algorithm over finite fields – Multiplication: map to exact floating point over medium-sized prime finite fields (ATLAS package) – Multiplication: SSE 128-bit operations on Intel/AMD processors for small finite fields – Multiplication: modular algorithm for matrices over integers – Multiplication: evaluation/interpolation algorithm for matrices over K[x] and K[x]/hf(x)i – Tensor products of matrices – Fast inverse of a rational matrix using modular methods – Fast algorithm for powering matrices over finite fields using the primary rational form – Order/projective order of matrices over finite fields (Leedham-Green algorithm) – Row and column operations – Submatrix operations – Fast trace of product of matrices

The Strassen algorithm for multiplying two n × n matrices over an arbitrary ring takes log 7 2.81 3 O(n 2 ) =∼ O(n ) arithmetic operations, instead of the classical O(n ) operations. Magma uses the Strassen algorithm at whichever dimension is applicable for the relevant ring. Magma Overview V2.18 (December 16, 2011) 45

8.1.3 Echelon Form and Related Operations – Echelonization over fields and euclidean domains – Nullspace (many different techniques including sparse, p−adic and modular algorithms) – Solution of systems of linear equations – Rank (for matrices over domains or Euclidean rings) – Determinant (for matrices over any commutative ring; modular algorithms over Z and Q) – Determinant (efficient elimination and minor expansion algorithm for multivariate polynomial rings with many variables) Fundamental to Magma’s fast linear algebra module is an asymptotically-fast echelon form algorithm for fields, which maps the problem to multiplication, thus taking advantage of all the fast multiplication algorithms above. Based on this, there are also asymptotically-fast algorithms for computing the nullspace, rank, or determinant of a matrix over a field. For matrices over the integers or rationals, there are asymptotically-fast algorithms for all of the above operations, which are based on modular and p-adic methods. 8.1.4 Canonical Forms – Generalized Jordan canonical form of matrices over fields – Rational and primary rational canonical forms of matrices over fields – Hermite and Smith forms for matrices over Euclidean domains (fast modular algorithm over the integers) – Characteristic polynomial, minimal polynomial, – Eigenvalues and eigenspaces (modular algorithms used over Z and Q) – LLL-reduction of matrices over Z. Over finite fields, a fast algorithm due to Allan Steel is used to construct the various matrix canonical forms: generalized Jordan, rational, and primary rational [40]. Over the ring of integers, asymptotically-fast modular algorithms are used to compute characteristic and minimal polynomials polynomials and the Hermite and Smith normal forms. Given a 100 × 100 matrix over Z with random one-digit entries Magma finds its Smith form in 0.06 seconds (the largest elementary divisor is 50 digits) and its characteristic polynomial in 0.1 seconds (on an 2.4GHz Opteron processor). 46 Magma Overview V2.18 (December 16, 2011)

8.2 Sparse Matrices A special type for sparse matrices is provided so that the user can build up such matrices and then apply some non-trivial algorithms to them. An extended example in the Handbook implements the basic linear sieve for discrete logarithms in the Magma language, thus demonstrating how one can use the sparse matrix facilities when implementing index-calculus methods. Features:

– Creation of sparse matrices in compact form. – Creation of trivial sparse matrices followed by dynamic expansion. – Basic properties (density, etc.). – Conversion between sparse and normal (dense-representation) matrices. – Multiplication of dense vectors by sparse matrices. – Non-trivial invariants of sparse matrices: nullspace, rank and elementary divisors (equivalent to Smith form). – Computation of non-zero solution vector for sparse systems arising in index-calculus algorithms (Structured Gaussian elimination and Lanczos algorithms [28]). – Computation of general nullspace over ﬁelds and Euclidean rings using Markowitz-pivoting techniques.

8.3 Vector Spaces

8.3.1 Construction

– Construction of vector spaces of n-tuples over a field – Construction of vector spaces comprising m × n matrices over a field – Extension and restriction of the field of scalars – Direct sum

8.3.2 Construction

– Vector arithmetic – Normalization, rotation – Tensor product of vectors – Trace of a vector in a subﬁeld – Weight, support

8.3.3 Subspaces and Quotient Spaces

– Construction of a subspace – Membership of a subspace – Transversal of a subspace (over a ﬁnite ﬁeld) – Complement of a subspace – Sum and intersection of subspaces – Reduction of vectors over a subspace – Quotient spaces Magma Overview V2.18 (December 16, 2011) 47

8.3.4 Bases – Construction of a vector space with speciﬁed basis – Coordinates of a vector with respect to a basis – Test for linear independence of a set of vectors – Extend a linearly independent set to a basis

8.3.5 Homomorphisms – Construction of Hom(U, V ), U and V vector spaces – Image, kernel cokernel – Echelon form

8.3.6 Quadratic Forms Every vector space is equipped with the standard inner product. Commencing with V2.7, the user may specify an arbitrary quadratic form.

– Creation of a vector space with a designated quadratic form (a quadratic space) – Inner product and norm of vectors

8.4 Free Modules In this section we are mainly concerned with free modules. We consider R-modules M of n-tuples where R has scalar action on M. The case in which the action of R on M is via a matrix algebra is considered in the section on Representation Theory.

8.4.1 Basic Operations – Construction of free modules of n-tuples – Construction of modules comprising m × n matrices – Arithmetic – Extension and restriction of the ring of scalars – Direct sum – Construction of submodules, quotient modules – Sum and intersection of submodules – Basis operations

8.4.2 Homomorphisms – Construction of modules Hom(M,N) for any free modules M and N – Explicit construction of Hom(U, V ) for proper subspaces U and V

– Explicit construction of Hom(H1,H2) for homomorphism modules H1 and H2 with left or right matrix action – Construction of the reduced module of a homomorphism module whose elements are with respect to the bases of the domain and codomain (not just the generic bases of these) – Image, kernel, cokernel – Echelon form (over a ﬁeld) – Hermite and Smith normal forms (over an ED) 48 Magma Overview V2.18 (December 16, 2011)

8.5 Modules over Dedekind domains Modules over dedekind domains are supported only for maximal orders of Algebraic number ﬁelds and Algebraic function ﬁelds.

– Creation of modules – Arithmetic with module elements – Submodules and quotients of modules – Determinant, dimension, pseudo-generators – Equality of modules, membership – Intersection of submodules – Product of a module by an ideal – Pseudo-basis and elementary divisors – Dual of a module – Steinitz class and Steinitz form – Construction of modules Hom(M,N) and standard calculations with morphisms

8.6 Chain Complexes Complexes of modules are a fundamental object in homological algebra. Conceptually, a complex is an inﬁnite sequence of modules, indexed by integers, with maps between successive modules such that the composition of any two maps is zero.

– Creation of a complex from a list of A-modules – Subcomplexes and quotient complexes – Operations on complexes: Splice, shift, direct sum – Exact extensions, zero extensions – Dual of a complex – Homology groups of a complex – Boundary maps – Construction of chain maps between complexes – Composition of chain maps – Image, kernel and cokernel of a chain map – Predicates for chain maps: Surjection, injection, isomorphism – Injective resolution (for modules over a basic algebra) – Projective resolution (for modules over a basic algebra) – Extending cohomology elements as chain maps – Maps induced on homology by chain maps, long exact homology sequence Magma Overview V2.18 (December 16, 2011) 49

9 Lattices and Quadratic Forms 9.1 Integral Lattices A lattice in Magma is a Z-module contained in Qn or Rn, together with a positive deﬁnite inner product. The information specifying a lattice is a basis, given by a sequence of elements in Zn, Qn or Rn, and a bilinear product (·, ·), given by (v, w) = vMwtr for a positive deﬁnite matrix M.

9.1.1 Constructions and Operations – Creation of a lattice by a given generating matrix or basis matrix together with an optional inner product matrix – Creation of a lattice by a given Gram matrix – Construction of lattices from codes – Construction of lattices from algebraic number ﬁelds

– Construction of special lattices, including the root lattices An, Dn, En; the laminated lattices Λn (including the Barnes-Wall lattice Λ16 and the Leech Lattice Λ24); the Kappa lattices Kn, etc. – Inner product, norm, and length of lattice elements with respect to the inner product on the lattice – Rank, determinant, basis, basis matrix, inner product matrix, Gram matrix, testing for integrality and evenness, index in a superlattice – Sum, intersection, direct sum, tensor product, exterior square, symmetric square of lattices – Creation of sublattices and superlattices, scaling of lattices – Creation of quotient lattices (abelian group with isomorphism) – Dual of a lattice, dual quotient of a lattice – Action on lattice elements by matrices

Several families of interesting lattices are directly accessible inside Magma using standard constructions, e.g., root lattices and preimages of linear codes. For each lattice, a LLL-reduced basis for the lattice is computed and stored internally and subsequently used for many operations.

9.1.2 Reduction Algorithms Central to the lattice machinery in Magma is a highly optimized LLL algorithm which constructs a basis of the lattice which is LLL-reduced. Usually the vectors of the new basis have very small norms. The default method for LLL reduction in Magma is based on the rigorous ﬂoating-point LLL algorithm of Nguyen and Stehl´e.

– LLL reduction of lattices using the Nguyen-Stehléfloating-point method. This is actually a family of algorithms that allow the user to choose the degree of reduction ranging from a guaranteed LLL basis to much weaker but faster reduction. – LLL reduction of lattices using the exact de Weger integral method – LLL reduction of lattices using the Schnorr-Euchner deep insertion strategy – HKZ (Hermite-Korkine-Zolotarev) reduction of lattices, a stronger form of reduction than LLL – Gauss reduction is available for 2-dimensional lattices. This is guaranteed to produce a basis whose vectors have norms which achieve the minimum – Seysen reduction of a lattice L, that is, the construction of a lattice L0 from L such that L0 and its dual are simultaneously reduced – Pairwise reduction of lattices, basis matrices and Gram matrices – Orthogonalization and orthonormalization (Cholesky decomposition) of a lattice – Testing matrices for being positive or negative (semi-)definite

The Nguyen-Stehléfloating-point method for LLL reduction can reduce matrices with very large entries as well as matrices having large sizes (e.g., number of rows well over 500). Indefinite Gram matrices may also be reduced in some cases. 50 Magma Overview V2.18 (December 16, 2011)

9.1.3 Properties Magma includes a highly optimized algorithm for enumerating all short vectors in a lattice with given norm. This algorithm, developed by Damien Stehl¨e,is used for computing the minimum, the shortest vectors, short vectors in a given range, and vectors close to or closest to a given vector (possibly) outside the lattice.

– Enumeration of all shortest vectors of a lattice – Enumeration of all short vectors of a lattice having norm in a given range – Enumeration of all vectors of a lattice having squared distance from a vector (possibly) outside the lattice in a given range – Enumeration of all vectors of a lattice closest to a vector (possibly) outside the lattice – Process to enumerate short or close vectors of a lattice thereby allowing manual looping over short vectors having norm in a given range or close vectors having squared distance in a given range – Minimum, packing radius, kissing number – Hermite constant, centre density, density – Successive minima, theta series – Pure lattice of a lattice over Z or Q – Construction of a fundamental Voronoi cell of a small-dimensional lattice – Holes, deep holes and covering radius of a lattice

As an example, the 98280 (normalized) shortest vectors of the Leech lattice Λ24 are constructed in 0.47 seconds. – Computation of the p-neighbour of a lattice with respect to a given vector and prime p – The sequence of p-neighbours a lattice at a prime p – The transitive closure of the p-neighbour relation of a lattice at a prime p – Spinor genus of an integral lattice, returned as a sequence of isometry representatives – Genus of an integral lattice, returned as a sequence of isometry representatives – Adjacency matrix of a sequence of lattices under the p-neighbour relation, for a sequence closed under the p-neighbour relation – p-Adic Jordan form for lattices

The genus of the 12-dimensional Coxeter-Todd lattice K12 is enumerated in 16 seconds and has 16 classes of lattices and mass 4649359/4213820620800 ≈ 0.000001103359. Magma Overview V2.18 (December 16, 2011) 51

9.2 Lattices with Group Action: G-Lattices In Magma, a G-lattice L is a lattice upon which a ﬁnite integral matrix group G acts by right multiplication. Magma provides support for computations with G-lattices. The most important operation is the calculation of the automorphism group of a lattice.

9.2.1 Automorphism Groups The computation of the automorphism group of a lattice (i.e. the largest matrix group that acts on the lattice) and the testing of lattices for isometry is performed within Magma using orthogonal decomposition (due to Gabi Nebe) and a backtrack search designed by Bill Unger in 2009. The backtrack search is based on the Plesken-Souvignier backtrack algorithm [PS97] together with ordered partition methods due to Leon.

– Automorphism group of a lattice – Subgroup of the automorphism group of a lattice fixing specified bilinear forms – Determination as to whether two lattices are isometric – Determination as to whether two lattices are isometric in such a way that specified bilinear forms are fixed – Determination as to whether a simultaneous isometry exists between two sequences of lattices – Determination as to whether a lattice has an isometric embedding into a second lattice

The automorphism group of the 24-dimensional Leech lattice Λ24 is found in 18.760 seconds. The group has order 8315553613086720000 and is the double cover of the sporadic simple group Co1.

9.2.2 G-Lattices If G is a finite integral matrix group, then Magma uses the Plesken centering algorithm ([Ple74]) to construct all G-invariant sublattices of a given G-lattice L. The lattice of G-invariant sublattices of L can be explored in much the same way as the lattice of submodules of an A-module, where A is an algebra over a finite field.

– Creation of G-lattices with associated operations – Invariant lattice of a rational matrix group and the associated action (thus yielding an integral representation of the group) – Bravais group of a finite rational matrix group – Space of invariant bilinear forms – Symmetric forms, antisymmetric forms, positive definite symmetric form – Endomorphism ring – Centre of the endomorphism ring – Dimension of the space of invariant bilinear forms, the endomorphism algebra or its centre using a modular algorithm – G-invariant sublattices of finite index of either a G-lattice or integral matrix group – A G-invariant sublattice of a lattice G can be created as a lattice of subsets with respect to inclusion. All the standard operations for working with this lattice are provided: e.g., meet, join, maximal sublattices, minimal overlattices. 52 Magma Overview V2.18 (December 16, 2011)

9.3 Quadratic Forms

9.3.1 General Quadratic Forms A quadratic form may be given either as a Magma lattice, as a multivariate polynomial or as a symmetric matrix. The major feature provided is Simon’s algorithm for ﬁnding isotropic subspaces of integral forms.

– Representation of a quadratic form either as a multivariate polynomial or as a symmetric matrix. – The p-signature of a form deﬁned over the rationals, for p a prime – The p-excess of a form deﬁned over the rationals, for p a prime

– The Witt (or Hasse-Minkowski) invariant over Qq. The form must be deﬁned over Z or Q. – Construction of an isotropic subspace using an algorithm due to Simon. The form must be deﬁned over Z or Q. In many cases the subspace returned will be guaranteed to be a maximal totally isotropic subspace.

9.3.2 Binary Quadratic Forms

Binary quadratic forms serve as a model for ideals in quadratic ﬁelds. Forms of negative discriminant are identiﬁed with lattices in the complex plane, which is the natural domain for modular forms and functions. Magma contains full functionality for forms of positive discriminant.

– Prime form, random form – Reduction of forms – Composition and powering – Enumeration of reduced forms and reduced orbits – Treatment of fundamental and nonfundamental discriminants – Class number and class group – Action of SL(2, Z) and PSL(2, Z) – Evaluation of modular functions on quadratic forms – Discrete logarithm of a form (for imaginary quadratic ﬁelds) Magma Overview V2.18 (December 16, 2011) 53

10 Associative Algebras

In this section we are mainly concerned with associative algebras – the facilities for Lie algebras and quantum algebras may be found in the section on Lie theory. The three chief ways of deﬁning algebras in Magma are in terms of a ﬁnite presentation, in terms of structure constants, or as a matrix (linear) algebra.

• Finitely presented associative algebras • General finite dimensional algebras (defined by structure constants) • Finite dimensional associative algebras (defined by structure constants) • Group algebras • Matrix algebras • Quaternion algebras • Basic algebras

10.1 Finitely Presented Associative Algebras Finitely-presented (FP) associative algebras (or noncommutative polynomial rings) are deﬁned by taking R-linear combinations of elements of a semigroup, where R is some ring. Since V2.11, these are handled by an extension of the commutative algebra machinery to noncommutative data structures and algorithms, where applicable. These include a noncommutative analogue for Gr¨obnerbases.

– Construction of free algebras over arbitrary fields – Arithmetic – Mappings into other associative algebras – Definition of left, right, two-sided ideals – Noncommutative Gröbnerbases of ideals, with specialized algorithms for different coefficient fields (fraction-free methods for the rational field and rational function fields) – Gröbnerbases of ideals over finite fields and rationals, using noncommutative extension of the Faugère F4 algorithm – Construction of degree-d (truncated) Gröbnerbases – Normal form of a polynomial with respect to an ideal – Construction of FP-algebras as quotient rings – Enumeration of the basis of finite-dimensional FP algebras – Matrix and structure-constant representations of finite-dimensional FP algebras – Construction of a matrix representation (Linton’s vector enumerator)

There are two major tools for computing with these algebras. The main approach is to apply a noncommutative version of Buchberger’s algorithm to construct a Gröbnerbasis for an ideal. This technique has been developed chiefly by Teo Mora in Genova and Ed Green in Virginia. An extension of Faugère’s F4 algorithm, due to Allan Steel, works by sparse linear algebra and is often much quicker. Linton’s vector enumerator uses the Todd-Coxeter algorithm in an attempt to construct a matrix representation. If the user has some idea as to how to select ideals that might give rise to matrix representations of reasonable degree, this approach is very successful. 54 Magma Overview V2.18 (December 16, 2011)

10.2 Exterior Algebras A special type is provided for working with an exterior algebra. Such an algebra is a special kind of FP algebra which is skew-commutative and is represented as a quotient of the free algebra 2 Khx1, . . . , xni by the relations xi = 0 and xixj = −xjxi for 1 ≤ i, j ≤ n, i 6= j. Because of the above relations, elements of an exterior algebra can be written in terms of commutative monomials in the variables (via a collection algorithm), and the associated algorithms are much more efficient than for the general noncommutative case. Gröbnerbases of ideals can be computed very efficiently (the Faugere F4 algorithm has been specially adapted for this). Furthermore, the extensive module theory also works over exterior algebras.

10.3 General Finite-Dimensional Algebras These algebras are presented in terms of a basis for a free module M together with a set of structure constants deﬁning the multiplication of these basis elements. It is assumed that we have an echelonization algorithm for M so that standard bases may be constructed for submodules.

– Creation of algebras in terms of structure constants – Direct sum – Arithmetic including Lie bracket operation – Identities: associative, commutative, Lie, etc – Properties of elements: idempotent, unit, zero-divisor, nilpotent – Trace and minimal polynomial – Creation of subalgebras, ideals and quotient algebras – Ideal arithmetic: Sum, product, powers, intersection – Ideal structure: Jacobson radical, maximal (minimal) left, right, two-sided ideals – Decomposition: Simplicity, semi-simplicity, composition series

10.4 Finite-Dimensional Associative Algebras These algebras are presented in terms of a basis for a free module M together with a set of structure constants deﬁning the multiplication of these basis elements. It is assumed that we have an echelonization algorithm for M so that standard bases may be constructed for submodules. We shall refer to these algebras as ASC-algebras.

10.4.1 Constructions and Element Operations

– Creation of algebras in terms of structure constants – Direct sum – Arithmetic including Lie bracket operation – Properties of elements: idempotent, unit, zero-divisor, nilpotent – Trace and minimal polynomial

10.4.2 Ideal and Subalgebra Structure

– Creation of subalgebras, ideals and quotient algebras – Ideal arithmetic: Sum, product, powers, intersection – Centralizer, idealizer – Characteristic ideals: Centre, commutator ideal, Jacobson radical – Ideal structure: Maximal (minimal) left, right, two-sided ideals Magma Overview V2.18 (December 16, 2011) 55

– Decomposition: Simplicity, semi-simplicity, composition series – Construction of the (left, right) regular matrix representation – Lie algebra deﬁned by the Lie product

Functions relating to the ideal structure (Jacobson radical, composition series, maximal and minimal ideals, etc) are implemented by applying the module theory machinery to the regular representation of the algebra.

10.4.3 Orders

– Construction of orders of algebras over the rationals or a number field – Construction of a maximal order of a central simple algebra defined over the rational numbers or a number field – Basis of an order – Construction of elements of an order of an algebra – Arithmetic of elements of an order of an algebra – Norm, trace, conjugate, minimal polynomial and representation matrix of elements – Construction of left, right and two-sided ideals of orders – Addition and multiplication of ideals – Left and right order of an ideal, colon ideal – Basis and basis matrix of an ideal 56 Magma Overview V2.18 (December 16, 2011)

10.5 Matrix Algebras Matrix algebras arise naturally as the endomorphism ring of a module and consequently are a fundamental structure in mathematics. Much effort has been invested into developing highly efficient code for working with these algebras and their elements in Magma. A description of generic matrix operations may be found in the section on Vector Spaces and Matrices and will not be repeated in this section. While a matrix algebra may be defined over any ring R, most non-trivial operations require R to be an Euclidean Domain.

10.5.1 Constructions and Element Operations

– Arithmetic – Extension and restriction of coefficient ring – Direct sum, tensor product, exterior square, symmetric square – Determinant (including modular algorithm), trace, characteristic polynomial, minimum polynomial – Order of a unit (Leedham-Green algorithm) – Canonical forms over a field: echelon, Jordan, rational, primary rational (asymptotically-fast Strassen- based algorithms over finite fields and modular algorithms over Q) – Canonical forms over an ED: echelon, Hermite, Smith (asymptotically-fast modular algorithms over Z) – Characteristic polynomial, minimal polynomial – Properties of an element: unit, zero-divisor, nilpotent

10.5.2 Ideal and Subalgebra Structure

– Standard basis for subalgebras, left, right and two-sided ideals – Quotient algebras – Sum, intersection, product, power of ideal – Radical of an ideal – Centre, commutator algebra – Centralizer of a subalgebra in the complete matrix algebra

– Jacobson radical (Brooksbank-O’Brien algorithm used when base ﬁeld is Fq)

– Unit group of an algebra over Fq (Brooksbank-O’Brien algorithm) – Maximal (minimal) left, right, two-sided ideals – Construction of the (left, right) regular matrix representation – Diagonalisation of a commutative algebra over a ﬁeld – Construction of Z-basis of a maximal order of a central simple algebra over Z

10.5.3 Presentations Algorithms have been developed by Carlson and Matthews which construct a presentation for a matrix algebra in terms of generators and relations. At present this and related machinery is restricted to algebras over ﬁnite ﬁelds. The ability to compute presentations is needed as part of the development of module theory over matrix algebras.

– Primitive idempotents – The condensed algebra eAe where e is a sum of primitive idempotents, one for each simple A-module – A presentation for the algebra in terms of generators and relations – The facility to write any element of the algebra as a word in the generators of the above presentation – Cartan matrix Magma Overview V2.18 (December 16, 2011) 57

10.6 Group Algebras A group algebra may be created for a ﬁnite group of moderate order over a Euclidean Domain.

– Creation of group algebras: a vector and term representation are provided allowing the construction of algebras for groups of arbitrary size. – Arithmetic including Lie bracket operation – Properties of elements: idempotent, unit, zero-divisor, nilpotent – Trace and minimal polynomial – Creation of subalgebras, ideals and quotient algebras – Ideal arithmetic: Sum, product, powers, intersection – Centralizer, idealizer – Augmentation ideal, augmentation map – Characteristic ideals: Centre, commutator ideal, Jacobson radical – Ideal structure: Maximal (minimal) left, right, two-sided ideals – Decomposition: Simplicity, semi-simplicity, composition series – Construction of the (left, right) regular matrix representation

10.7 Quaternion Algebras A quaternion algebra is a central, simple algebra of dimension four over a field. Basic functions are provided for quaternion algebras over an arbitrary field. Higher level routines are available for algebras over Q, number fields, and rational function fields k(x), where k is a finite field. This includes support for orders and ideals, in particular enumeration of left, right and two-sided ideal classes. – Arithmetic of elements – Norm, trace, and conjugation – Minimal polynomial of elements – Discriminant, ramified primes and ramified places – Construction of maximal orders – Recognition of quaternion algebras – Computation of an isomorphism to the matrix algebra, in particular over completions of the base field – Embedding subfields into quaternion algebras – Left and right orders of ideals – Isomorphism testing for ideals and orders – Construction and recognition of Eichler orders – Enumeration of left, right and two-sided ideal classes of Eichler orders – Enumeration of conjugacy classes of orders – Unit groups of orders in definite algebras 58 Magma Overview V2.18 (December 16, 2011)

10.8 Basic Algebras A basic algebra is a finite dimensional algebra A over a field, all of whose simple modules have dimension one. In the literature such an algebra is known as a “split” basic algebra. Every finite dimensional algebra is Morita equivalent to a basic algebra, meaing that the algebra and its basic algebra have equivalent module categories and hence the same representation theory. So, for example, a cohomology calculation involving modules over an algebra is often most easily done by condensing to the basic algebra. Magma has the capabilility of constructing the basic algebra of a matrix algebra defined over a finite field. In some cases it is necessary to extend the field in order to split the irreducible modules over the matrix algebra. The basic algebra type in Magma is optimized for the purposes of doing homological calculations.

10.8.1 Construction

– Creation from a sequence of projective modules and a path tree for each module – Creation of the split basic algebra over a finite field corresponding to a matrix algebra or endomorphism algebra. – Creation of the basic algebra corresponding to a Schur algebra or Hecke algebra over a finite field. – Creation of the basic algebra corresponding to the group algebra of a p-group over GF (p). – Arithmetic – Extension and restriction of the coefficient ring – Tensor product – Opposite algebra

10.8.2 Modules and Cohomology

– Construction of modules over basic algebras – Submodules, quotient modules, radicals and socles – Algebra considered as a right regular module over itself – The space Hom(M,N) of all homomorphisms (all projective homomorphisms) from module M to module N – Pushouts and pullbacks with respect to module homomorphisms – Projective resolution as a complex of modules; projective covers – Injective resolution as a complex of modules; injective hulls – Calculation of the Ext algebra of a basic algebra – Restriction and inﬂation for basic algebras of p-groups – Cohomology ring of the unique simple module k for the basic algebra of a p-group

– Calculation of A∞ algebras structures on cohomology rings Magma Overview V2.18 (December 16, 2011) 59

11 Representation Theory

This section describes facilities in Magma that relate to the representation theory of groups and associative algebras. The main topics considered include:

• Modules over an algebra • K[G]-modules • Representations of groups • Character theory • Invariant theory

11.1 Modules over an Algebra We consider a module whose elements are n-tuples over a field K with an action given by a matrix representation of an associative algebra A. We will refer to these modules as A-modules. These include K[G]-modules. The four fundamental algorithms for computational module theory are echelonization, the spinning algorithm, the meataxe algorithm and an algorithm for Hom(U, V ). For the important case of modules over finite fields, different representations of vector arithmetic, depending upon the field, have been implemented.

11.1.1 Creation

– Creation from the matrix representation of an associative algebra. – Creation from group actions of diﬀerent kinds – Permutation module of a group corresponding to its action on the cosets of a subgroup – K[G]-modules corresponding to actions of a permutation or matrix group on a polynomial ring.

11.1.2 Constructions

– Extension and restriction of the field of scalars – Direct sum – Tensor product, symmetric square, exterior square (K[G]-modules only) – Dual (K[G]-modules only) – Induction and restriction (K[G]-modules only) – All irreducible K[G]-modules of a finite soluble group where K is a finite field or field of characteristic zero – All irreducible K[G]-modules of a finite group where K is restricted to be a finite field or the rational field.

11.1.3 Submodules and Quotient Modules

– Submodules via the spinning algorithm – Membership of a submodule – Basis operations – Sum and intersection of submodules – Quotient modules 60 Magma Overview V2.18 (December 16, 2011)

11.1.4 Structure

– Splitting a reducible module (Holt-Rees Meataxe) – Testing a module for irreducibility, absolute irreducibility – Centralizing algebra of an irreducible module – Composition series, composition factors, constituents – Maximal and minimal submodules – Jacobson radical, socle – Socle series – Existence of a complement of a submodule – One complement, all complements of a direct summand – Testing modules for indecomposability; indecomposable components – Submodule lattice for modules over a ﬁnite ﬁeld

The Magma algorithm for splitting modules (the Meataxe algorithm) is a deterministic version of the Holt-Rees algorithm and is capable of splitting modules over GF(2) having dimension up to at least 20 000. Since V2.16, a new Meataxe algorithm is used for splitting general A-modules, where A is a finite dimensional matrix algebra defined over the rational field. This yields an effective algorithm for decomposing a module into indecomposable summands. If the module is a G-module for some group G, extensive use is also made of character theory. Representations associated with characters having non-trivial Schur indices are properly handled. The difficult problem of splitting homogeneous modules (direct sums of the same indecomposable) is handled by decomposing the endomorphism ring of the module via a maximal order. Modules having dimensions in the several hundreds are routinely split into indecomposable modules.

11.1.5 Homomorphisms

– Construction of Hom(U, V ), U and VR-modules – Endomorphism ring of a module – Automorphism group of a module – Testing modules for isomorphism

Magma includes a new algorithm for the construction of Hom(U, V ) which is applicable to modules having dimension several hundred.

11.2 Ordinary Representations

11.2.1 Splitting of G-modules

11.2.2 Construction of Irreducible G-modules

– Dixon’s method to compute the representation from a character – For soluble groups: Brückner’s method to compute all absolutely irreducible representations – For general finite groups: a new algorithm of Steel constructs all irreducible representations over the rational field Magma Overview V2.18 (December 16, 2011) 61

11.2.3 Change of Ring – Given an absolutely irreducible G-module over a number field or finite field, write it over any related field possible. Find a smallest field of definition – Restriction of scalars: use the module structure of the coefficient ring to obtain reducible representations over any subfield – For representaions over Q, find isomorphic integral representations – For representations over number fields, decide if the representation can be made integral. Find all classes of integrally equivalent representations of an absolutely irreducible one. – Compute modular representations from representations over number fields at any prime ideal – Try to find “nicer” versions of a representation

11.2.4 Properties – Compute the character of the representation – Decide (absolute) irreducibility – Compute forms invariant under the action – Compute modules invariant under the action

11.3 Representations of Symmetric Groups Special functionality for representations of a symmetric group concentrates on characters as indexed by partitions of weight the degree of the group.

– Integral, seminormal and orthogonal representations of a permutation. – Values of a character of a symmetric group indexed by a partition on a permutation. – Characters of symmetric groups corresponding to partitions. – Values of a character of an alternating group indexed by a partition on a permutation. – Characters of symmetric groups corresponding to partitions.

11.4 Character Theory The character theory machinery is currently restricted to characters deﬁned over the complex ﬁeld.

– Deﬁnition of class functions – Construction of permutation characters – Arithmetic on class functions: sum, diﬀerence, tensor product – Frobenius-Schur indicator – Norm, order, kernel, centre of a character – Properties: generalized character, character, irreducible, faithful, linear – Induction and restriction of a character – Decomposition of a tensor power: orthogonal components, symmetric components – Action of a group on the characters of a normal subgroup – Decomposition of characters – Class matrix, structure constants for centre of group algebra – Table of ordinary irreducible characters (Dixon-Schneider algorithm, Unger’s algorithm) 62 Magma Overview V2.18 (December 16, 2011)

12 Lie Theory

The current elements of the machinery for Lie theory comprise:

• Coxeter systems • Root systems • Root data • Coxeter groups • Reﬂection groups • Finite dimensional Lie algebras • Quantized enveloping algebras (aka quantum groups) • Groups of Lie type

12.1 Coxeter systems

– The standard descriptions for Coxeter systems and reflection groups are all supported: Coxeter matrices, Coxeter graphs, Cartan matrices, Dynkin digraphs, and Cartan names. – Conversion between the different descriptions. – Testing isomorphism and Cartan equivalence of Coxeter systems. – Testing for properties such as finite, affine, hyperbolic, and compact hyperbolic Coxeter systems. – Construction of any finite, affine and hyperbolic system. – Determining the size and number of roots of (finite) Coxeter systems. – Dynkin diagrams for finite systems. – Predicates: IsIrreducible, IsCrystallographic, IsSimplyLaced.

12.2 Root Systems A root system describes the reflections in a reflection group and plays an essential role in the theory of finite Coxeter groups and Lie algebras.

12.2.1 Creating Root Systems

– Any ﬁnite root system can be constructed by giving its simple roots and coroots, including non- semisimple systems (where the dimension of the vector space is larger than the rank). – Semisimple root systems may be constructed from a Coxeter matrix, Coxeter graph, Cartan matrix, Dynkin digraph, or Cartan name. – Standard root systems may be constructed – these are systems whose pairing is the Coxeter form, and is the way in which root systems are frequently given in the literature. – Direct sums and duals of root systems are supported.

12.2.2 Operations and Properties

– Test equality, isomorphism and Cartan equivalence. – Determine any of the following descriptions for a root system: Cartan name, Coxeter diagram, Dynkin diagram, Coxeter matrix, Coxeter graph, Cartan matrix, or Dynkin digraph. – Invariants such as base ﬁeld, rank, dimension, Coxeter group order. – Properties such being irreducible, semisimple or simply laced. Magma Overview V2.18 (December 16, 2011) 63

12.2.3 Roots and Coroots The (co)roots are stored in an indexed set, with positive roots first. The can be described and manipulated via their index, or as vectors with respect to either the standard basis or the basis of simple (co)roots. – Root space and coroot space. – Construction of the complete set of roots or coroots. – Conversion between indices and vectors. – Highest long or short root. – Reflection actions of the (co)roots: given as matrices, permutations, or words in the simple reflections. – Basic arithmetic with (co)root indices: sum, negation, positivity, heights, norms. – Coxeter form and dual Coxeter form.

12.3 Root Data Root data are fundamental to the study of Lie algebras and groups of Lie type whereas the closely related concept of a root system discussed above is normally used when working with Coxeter groups or reﬂection groups.

12.3.1 Constructions – A split (untwisted) root datum can be constructed by giving its simple roots and coroots. – A semisimple system may be constructed from a Cartan matrix, Dynkin digraph, or Cartan name. By default the adjoint datum is returned, but the isogeny type can be speciﬁed. – Standard root systems may be constructed – these are systems whose pairing is the Coxeter form, and is the way in which root systems are frequently given in the literature. – Constructions are provided for direct sums, duals and subdata of root data.

12.3.2 Operations and Properties – Equality, isomorphism, Cartan equivalence, and isogeny. – Determination any of the following descriptions for a root datum: Cartan name, Coxeter diagram, Dynkin diagram, Coxeter matrix, Coxeter graph, Cartan matrix, Dynkin digraph – Elementary invariants: Base ﬁeld, rank, dimension, Coxeter group order, group of Lie type order. – Determination of the fundamental and (co)isogeny groups. – Determination of properties such as being irreducible, semisimple, crystallographic, simply laced, adjoint and simply connected. – The standard constants used to deﬁne Lie algebras and groups of Lie type can be computed: p, q, N, ², M, C, and eta.

12.3.3 Roots, Coroots and Weights The (co)roots are stored in an indexed set, with positive roots ﬁrst. The can be described and manipulated via their index, or as vectors with respect to the standard basis or the basis of simple (co)roots or the basis of fundamental weights. – Root space and coroot space. – Construction of the complete set of roots or coroots. – Conversion between indices and vectors. – Highest long or short root. 64 Magma Overview V2.18 (December 16, 2011)

– Reﬂection actions of the (co)roots: given as matrices, permutations, or words in the simple reﬂec- tions. – Basic arithmetic with (co)root indices: sum, negation, positivity, heights, norms. – Left and right strings through one root in the direction of another. – Coxeter form and dual Coxeter form. – (Co)weight lattice and fundamental (co)weights

12.4 Coxeter Groups 12.4.1 General Coxeter Groups as FP-Groups General Coxeter groups are implemented as a subclass of finitely presented groups so that they inherit all the operations for finitely presented groups as well as having many specialized functions. The main difference is that every word is automatically converted into normal form using an algorithm designed and implemented by Bob Howlett. This module was implemented by Bob Howlett, Scott Murray, and Don Taylor.

– A Coxeter group can be constructed from a Cartan matrix, Dynkin digraph, Cartan name, root system, or root datum. – Test isomorphism of Coxeter groups. – Elementary operaions include determining the Cartan name, Coxeter diagram, Coxeter matrix, Coxeter graph, rank. – Determination of basic properties such as being finite, affine, hyperbolic, compact hyperbolic, irreducible or simply laced. – Arithmetic of words: identity, multiplications, inversion, powers. – Degrees of the basic invariant polynomials. – Coxeter element and Coxeter number. – Braid group and pure braid group – Conversion to and from permutation and reflection representations. – Construction of the standard parabolic subgroups – The growth function of a Coxeter group may be computed using a very fast algorithm due to R. Howlett.

12.4.2 Finite Coxeter Groups as Permutation Groups

– A permutation Coxeter group can be constructed from a Cartan matrix, Dynkin digraph, Cartan name, root system, or root datum. – Finite Coxeter groups are implemented as a subclass of permutation groups so that they inherit all the operations for permutation groups. – In addition to the standard functions for groups, almost all of the functions for root systems and root data also apply to permutation Coxeter groups. – A reflection subgroup can be represented two ways: As a permutation group on the roots of the larger groups, or as a permutation group on its own roots. – Transversals of reflection subgroups may be computed using an efficient algorithm due to Don Taylor. – The “standard” permutation action of a Coxeter group (usually the smallest degree permutation action) may be computed. For example, the standard action of the group of type An, gives the symmetric group on n + 1 points. Magma Overview V2.18 (December 16, 2011) 65

12.5 Complex Reﬂection Groups

– Construction and identification of a reflection group over an arbitrary ring, given the simple roots, coroots and orders – Construction of real reflection groups from a Cartan matrix, Dynkin digraph, Cartan name, root system, or root datum – Construction of all finite complex reflection groups – The degrees of the fundamental invariants may be computed for any complex reflection group. Basic codegrees can also be computed. – Most of the functions available for Coxeter groups are also available for real reflection groups.

12.6 Finite-Dimensional Lie Algebras A finite-dimensional Lie algebra L over a field K is presented in terms of a basis for a K-vector space V together with a set of structure constants defining the multiplication of these basis elements. The major structural machinery for Lie algebras has been implemented for Magma by Willem de Graaf.

12.6.1 Construction and Arithmetic

– Creation of Lie algebras in terms of structure constants – Construction of a Lie algebra from an associative algebra via the Lie bracket product – Construction of a Lie algebra given by generators and relations – Construction of a Lie algebra from a p-group, by using its Jennings series. – Construction of a speciﬁed simple Lie algebra – Direct sum – Arithmetic – Trace and minimal polynomial

12.6.2 Properties and Invariants

– Test for abelian, nilpotent, solvable, restricted – Test for simple, semisimple – Killing form – Adjoint representation of an element; Associated adjoint algebra – Root system of a semisimple Lie algebra with a split Cartan subalgebra

12.6.3 Arithmetic of Subalgebras and Ideals

– Creation of subalgebras, ideals and quotient algebras – Ideal arithmetic: Sum, product, powers, intersection – Centre – Centralizer, normalizer – Jacobson radical, nil radical, solvable radical – Given a Lie algebra L deﬁned over a ﬁeld of characteristic p > 0, construction of the Lie subalgebra M of L generated by any set of elements of L. Thus, M is closed under the restriction map. 66 Magma Overview V2.18 (December 16, 2011)

12.6.4 Structure – Composition series – Derived series, lower central series, upper central series – Nilradical, solvable radical – Cartan subalgebra, Levi subalgebra – Maximal (minimal) left, right, two-sided ideals – Decomposition of a Lie algebra into a direct sum of ideals – Type of a simple or semisimple algebra

12.6.5 Representations – Construction of a faithful module over a Lie algebra of characteristic zero – Construction of highest-weight modules over split semisimple Lie algebras – Construction of tensor products, symmetric powers, antisymmetric powers of Lie algebra modules

12.6.6 Universal enveloping algebras – Construction of a universal enveloping algebra of a Lie algebra – A special construction of the universal enveloping algebra of a split semisimple Lie algebra, via a Kostant basis

12.6.7 Finitely Presented Lie Algebras – Construction of a Gröbnerbasis for a finitely presented (FP) Lie algebra – Construction of a Lie algebra with structure constants from an FP Lie algebra when finite-dimensional – Construction of a nilpotent quotient of an FP Lie algebra to a designated class

12.7 Quantized Enveloping Algebras A quantized enveloping algebra (corresponding to a given root datum) is represented with respect to an integral basis, as deﬁned by Lusztig.

– Constructing of quantized enveloping algebras with respect to a given root datum – Arithmetic: sum and product – Representations: construction of highest-weight modules, and tensor products of them – Construction of the canonical basis of a highest-weight module – Construction of elements of the canonical basis of the negative part of a quantized enveloping algebra – Action of the Kashiwara operators – Littelmann’s path model: action of the path operators, construction of the crystal graph

12.8 Groups of Lie Type Machinery is provided which allows computation in split (untwisted) groups of Lie type with the Steinberg presentation. These groups can be deﬁned over any Magma ﬁeld. Elements can be normalised using the Bruhat decomposition.

12.8.1 Creating Groups of Lie type A group of Lie type can be created from a ﬁeld and a Cartan name, Weyl group, root datum, Cartan matrix or Dynkin digraph. Magma Overview V2.18 (December 16, 2011) 67

12.8.2 Operations and Properties – Most of the operations and properties for root data also apply to groups of Lie type. – Equality, algebraic isomorphism, isogeny. – Algebraic group generators; abstract group generators for certain ﬁelds – Element arithmetic and normalisation – Bruhat decomposition and multiplicative Jordan decomposition. – The order of a twisted ﬁnite group of Lie type can be computed.

12.8.3 Automorphisms The inner, diagram, diagonal and ﬁeld automorphisms can be constructed. These include all algebraic group automorphisms, and in many cases all abstract group automorphisms.

12.8.4 Representation Theory – Standard, regular and highest weight representations can be constructed. – The inverse image of a module with respect to a given representation can be computed using a generalised row reduction function. The following operations exploit the bijection between modules of connected Lie groups and their highest weights. Their Magma implementation closely follows that in the Lie package of Cohen et al. – Convert between highest weight, dominant weights, or all weights of such a module, – Determine the dimension of a module. – Apply the Adams operator or the Demazure operator. – Compute plethysms, – Compute symmetric, alternating, Littlewood-Richardson, or regular tensor products, – Branch to a subgroup, or collect to a supergroup, and – Compute Kazhdan-Lusztig polynomials and R-polynomials.

12.9 Finite Groups of Lie Type The functions in this section use the theory of FGLT to determine the required information and consequently are applicable to groups far larger than those that can be handled by the generic matrix group machinery. – Functions allow the construction of generators for any FGLT in its natural representation. – The order of any ordinary or twisted FGLT can be computed. – The Sylow subgroups of any classical FGLT are contructed using an algorithm of Holt and Stather. – The conjugacy classes of elements of most of the classical FGLTs can be determined. – Likewise the conjugacy of any two elements can be determined. 68 Magma Overview V2.18 (December 16, 2011)

13 Commutative Algebra

The Magma facility for commutative rings allows the user to deﬁne any ring, starting from the ring of integers, by repeatedly applying the four basic constructions: transcendental extension, quotient by an ideal, localization, and completion. Rings derived from a polynomial ring will be considered in this section, while ﬁelds, their orders and valuation rings will be presented in the following section. The following rings and modules are considered here:

• Multivariate polynomial rings • Ideal theory of multivariate polynomial rings • Affine algebras • Modules over affine algebras • Localization in an affine algebra • Boolean Polynomial Rings

The basic computational problems for commutative rings include: – A canonical form for elements – Eﬃcient arithmetic – A canonical representation (i.e., standard basis) for ideals – Arithmetic with ideals – Formation of quotient rings – Localization at the origin – Ideal decomposition, i.e., primary decomposition – The study of modules over rings The fundamental tools on which most machinery for computational (commutative) ring theory is based include factorization of elements in a UFD, the eﬃcient construction of standard bases for ideals and the factorization of ideals. Magma Overview V2.18 (December 16, 2011) 69

13.1 Multivariate Polynomial Rings Multivariate polynomial rings in any number of variables may be formed over any coefficient ring, including a polynomial ring. Multivariate polynomials are represented in distributive form, using ordered arrays of coefficient-monomial pairs. Different orderings are allowed on the monomials; these become significant in the construction of Gröbnerbases of ideals. Computations with ideals are available (since V2.8) for ideals defined over general Euclidean rings as well as for ideals defined over fields.

13.1.1 Creation and Ring Operations

– Creation of a polynomial ring – Monomial orders: lexicographical, graded lexicographical, graded reverse lexicographical, block elimination, general weight vectors, etc. – Dynamic data structures for monomials providing optimal packing and rigorous detection of overflow – Base extend – Definition of a ring map – Kernel of a ring map – Properties of ring maps: Surjective, bijective Since V2.7, the original linked-list representation of polynomials has been replaced with a more compact random-access array structure, resulting in less memory usage and faster access. A new fraction-free representation for polynomials at the lowest level gives very significant speedups for arithmetic over some fields (particularly the rational field and rational function fields). A new representation employing variable byte sizes for monomials is also introduced in V2.7, requiring less memory and providing greater speed. The maximum total degree of any monomial has been increased to 230 − 1 = 1073741823. Monomial overflow is rigorously detected.

13.1.2 Arithmetic

– Arithmetic with elements – Fast multiplication and divison using heap-based algorithms of Monagan and Pearce – Fast powering in characterstic p via Frobenius map – Recursive coeﬃcient, monomial, term, and degree access – Diﬀerentiation, integration – Evaluation and interpolation – Properties: a unit, a zero-divisor, nilpotent

13.1.3 GCD and Factorization

– Greatest common divisor (sparse EEZ-GCD, fast GCD-HEU, evaluation-interpolation algorithms) – Newton polygon – Squarefree factorization – Bivariate factorization over GF(q), Z and Q (polynomial-time trace-based algorithm of Belabas et al., interpolation algorithms) – Multivariate factorization over GF(q), Z and Q (reducing to bivariate factorization to solve combination problem) – Factorization over arbitrary algebraic function ﬁelds (including inseparable ﬁeld extensions) – Resultant (modular and sub-resultant algorithms), discriminant 70 Magma Overview V2.18 (December 16, 2011)

Factorization of bivariate polynomials over all supported rings is accomplished by an algorithm which extends van Hoeij’s knapsack ideas for Z[x] to solve the hard combination problem for GF(q)[x, y][3]. The new algorithm runs in polynomial time and performs extremely well in practice. General multivariate factorization is reduced to this new bivariate algorithm, so a combination problem never arises for any number of variables. Shoup’s tree Hensel lifting algorithm has also been adapted for power series, making the lifting stages of all kinds of bivariate/multivariate factorization much faster than previously. Factorization over general algebraic function ﬁelds of small characteristic is accomplished by an algorithm of Allan Steel [43]. This can handle extensions which are inseparable, and may have an arbitrary number of both algebraic and transcendental generators. Resultants are computed using asymptotically-fast modular and evaluation/interpolation algorithms. Options are provided for Monte Carlo-style stopping on stability, which greatly speeds up the computation (since the bounds are usually very much worse than the required lifting level).

13.1.4 Gr¨obnerBasis

– Construction of ideals and subrings – Gröbnerbases of ideals over fields, with specialized algorithms for different coefficient fields (fraction- free methods for the rational field and rational function fields)

– Gröbnerbases of ideals over finite fields and rationals, using optimized Faugère F4 algorithm – Gröbnerbases of ideals over general Euclidean rings, using an extension by Allan Steel of Faugère’s F4 algorithm (includes integer ring Z, residue class rings Z/mZ, K[x]/hf(x)i [m, f arbitrary non- zero moduli] and Galois rings) – GröbnerWalk algorithm for converting the Gröbnerbasis of an ideal over a field from one monomial order to another order – Construction of ideals of boolean polynomial rings and Gröbnerbases of such ideals – FGLM algorithm for converting the Gröbnerbasis of a zero-dimensional ideal over a field from one monomial order to a different order (a fast p-adic method is used in the case of the rational field) – Construction of degree-d (truncated) Gröbnerbases – Construction of ideal via fixed basis (so as to determine coordinates of elements with respect to this basis) – Normal form of a polynomial with respect to an ideal – Reduction of ideal bases – S-polynomial of two polynomials

Since V2.11, an optimized implementation of Faugère’s F4 algorithm (which uses sparse linear algebra) to compute Gröbnerbases is available for ideals over finite fields and the rationals. See [45] for detailed timings. The first HFE cryptosystem challenge of J. Patarin is solved by Magma 2.8 hours and 14GB on an 2.4GHz Intel Xeon64. This involves solving a system of 80 quadratic equations in 80 variables over GF(2) (each input polynomial has about 1600 terms). The challenge was first solved by J.- C. Faugèrein 52 hours on an 1GHz Alpha in 2002 using his unpublished F5/2 algorithm (allowing for processor speed, Magma is about 3 times faster). At the time of writing, there is still no publically-available software besides Magma which can solve the challenge.

13.1.5 Arithmetic with Ideals

– Sum, product, intersection, colon ideal, – Saturation of an ideal, leading monomial ideal – Elimination ideals – Determination of whether a polynomial is in an ideal or its radical Magma Overview V2.18 (December 16, 2011) 71

– Properties of an ideal: Zero, principal, proper, zero-dimensional – Extension and contraction of ideals – Variable extension of ideals – Noether normalisation of ideals – Minimal bases for homogeneous ideals – Normalisation of the aﬃne quotient algebra of an ideal – Maximal regular sequences in polynomial ideals – Rees ideal of an ideal in a polynomial ring or aﬃne algebra – Milnor numbers and Tjurina numbers

13.1.6 Invariants for Ideals – Dimension and maximally independent sets – Hilbert series and Hilbert polynomial – Primary decomposition of an ideal – Triangular decomposition of a zero-dimensional ideal (algorithm of D. Lazard). – Probabilistic prime decomposition of the radical of an ideal – Equidimensional decomposition of an ideal – Radical of an ideal – Computation of the variety of a zero-dimensional ideal – Relation ideals (determination of algebraic relations between polynomials) – Syzygy modules – Construction of a minimal generating set of a polynomial subalgebra – Computations with polynomial generators of a submodule over a subalgebra in a polynomial ring Primary decomposition of ideals over general algebraic function ﬁelds of small characteristic is handled by a new algorithm of Allan Steel [43].

13.1.7 Gradings – Construction of graded polynomial rings with specific weights on the variables – Special efficient graded reverse lexicographical order with weights for optimal Gröbnerbasis of ideals in graded rings – All monomials of specific total or weighted degree – Homogeneous components of polynomials – Homogenization and dehomogenization of an ideal – Hilbert-driven Buchberger algorithm for fast computation of the Gröbnerbasis of a homogeneous ideal when the Hilbert series is known (using the F4 algorithm when possible) – Construction of degree-d (truncated) Gröbnerbases (respecting the grading of the polynomial ring)

13.2 Boolean Polynomial Rings Boolean polynomial rings provide efficient Gröbnerbasis computations with ideals in polynomial rings over GF(2) for which one desires to solve the underlying system of equations over GF(2). The effect of computing with an ideal in such a ring is the automatic inclusion of the field polynomials 2 xi + xi in the ideal. This saves time and memory because a compact bit vector representation is available for the monomials and conversion to and from this is not needed. – Construction of boolean polynomial rings with specific numbers of variables – Monomial orders: lexicographical, graded lexicographical, graded reverse lexicographical – Efficient construction of large polynomials in compact bit vector representation (via integers) – Gröbnerbases using the bit vector representation – Varieties 72 Magma Overview V2.18 (December 16, 2011)

13.3 Aﬃne Algebras

Let K be a ﬁeld, R = K[x1, . . . , xn] a polynomial ring over K and I an ideal of R. The quotient ring A = R/I is called an aﬃne algebra.

13.3.1 Creation and Operations – Creation of an aﬃne algebra – Arithmetic with elements

13.3.2 Arithmetic with Ideals

– Construction of ideals and subrings – Gröbnerbases of ideals – Ideal arithmetic: addition, multiplication, powers, quotients, colon ideals, intersections – Membership test for ideals – Test equality and inclusion of ideals – Construction of Rees ideals Affine algebras arise commonly in commutative algebra and algebraic geometry. They can also be viewed as generalizations of number fields and algebraic function fields. If the ideal J of relations defining an affine algebra A = K[x1, . . . , xn]/J is maximal, then A is a field and may be used with any algorithms in Magma which work over fields. Factorization of polynomials over such affine algebras is also supported. If an affine algebra has finite dimension considered as a vector space over the coefficient field, extra special operations are available on its elements.

13.4 Modules over Multivariate Rings

Modules over a multivariate polynomial ring R[x1, . . . , xn](R a Euclidean ring or ﬁeld) and quotient rings of such (aﬃne algebras) Magma has a special category for modules over three kinds of commutative multivariate rings:

• Multivariate polynomial rings.

• Localizations of Multivariate polynomial rings.

• Aﬃne Algebras.

Such rings are not principal ideal rings in general, so the standard matrix echelonization algorithms are not applicable. Magma allows computations in modules over such rings by adding a column ﬁeld to each monomial of a polynomial and then by using the ideal machinery based on Gr¨obnerbases.

13.4.1 Creation and Operations – Construction of modules with various module orders – Construction of graded modules with weights on the columns – Flexible choice of monomial orders (top-over-position, position-over-top, Schreyer order) – Arithmetic with elements – Construction of Gr¨obnerbases of modules – Row and column operations on elements – Tensor products Magma Overview V2.18 (December 16, 2011) 73

13.4.2 Submodules – Construction of submodules and quotient modules – Membership testing – Tensor product – Hilbert series of (homogeneous) modules – Multiplicity – Submodule sum, intersection, colon operation – Minimal bases for homogeneous modules – Multiplicity of a submodule

13.4.3 Homology – Syzygy modules – Construction of the image and kernel of a homomorphism – Hom(M,N), where M and N are modules

– Free resolutions, minimal free resolutions (Hybrid La Scala/F4 algorithm, iterative syzygy algorithm) – Betti numbers, homological dimension – Fitting ideals – Homology of a complex – Ext and T or of a complex – Dimension of cohomology groups of Serre twists of graded modules (Decker-Eisenbud-Floystad- Schreyer algorithm) Fundamental to Homology computations is a fast algorithm for computing free resolutions. Magma contains a hybrid of the algorithm of La Scala and the Faugere F4 Gr¨obnerbasis algorithm. This hybrid algorithm is able to compute resolutions of many modules in much faster time than for previous methods. 74 Magma Overview V2.18 (December 16, 2011)

13.5 Invariants for Groups Magma contains a substantial package for computing with invariant rings and fields of finite groups and algebraic groups, which been developed by Gregor Kemper and Allan Steel [24]. For finite groups, Magma supports computation in invariant rings over ground fields of arbitrary characteristic. Of particular interest is the modular case, i.e., the case where the characteristic of the ground field divides the order of the group. Magma includes an algorithm for computing primary invariants that guarantees that the degrees of the invariants constructed are optimal (with respect to their product and their sum) [25]. Beginning with V2.14, Magma also provides algorithms for finding invariants of linear algebraic groups. In particular, Derksen’s algorithm [15] and the algorithm by Beth and Müller-Quade[34] are included.

13.5.1 Construction of Primary and Secondary Invariants – Permutation and matrix group actions on polynomials – Independent homogeneous invariants of a speciﬁc degree – Molien series – Primary invariants having optimal degrees (with respect to their product and then sum) – Secondary invariants of optimal degrees (using a new algorithm for the modular case)

For the 4-dimensional representation of A5 over F2, optimal primary invariants (of degrees 3, 5, 8 and 12) are found in 1.5 seconds. For the cyclic matrix group of order 8 generated by the 5-dimensional Jordan form over F2, optimal primary invariants (of degrees 1, 2, 2, 4 and 8) are found in 0.6 seconds and secondary invariants with respect to these (of degrees 0, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 9, 9, 10 and 11) are found 14.4 seconds. This last computation took many hours with algorithms prior to those implemented in Magma V2.2.

13.5.2 The Ring of Invariants – Eﬃcient construction of fundamental invariants (direct and King algorithms) – Invariant ring as a graded module over the algebra generated by the primary invariants and explicit construction of the isomorphism – Invariant ring as a polynomial algebra – Determination of algebraic relations between secondary invariants – Module syzygies between secondary invariants – Algebraic relations between invariants

The algorithm of King for computing fundamental invariants is in general an order of magnitude faster than the previous direct method (which combined primaries and secondaries and reduces them to a minimal set). Using the King algorithm, fundamental invariants can be computed (each modulo a prime coprime with the group order) for all transitive groups of degree 7 in about 1 second total, and for all transitive groups of degree 8 in about 2 minutes total.

13.5.3 Properties – Hilbert series – Free resolution – Depth and homological dimension – Attribute control of invariant ring information – Determine whether an invariant ring is a polynomial ring or a Cohen-Macaulay ring Magma Overview V2.18 (December 16, 2011) 75

13.5.4 Invariants of Linear Algebraic Groups A linear algebraic group is an affine variety G together with morphisms giving G the structure of a group and is defined in Magma by simply giving polynomials defining G as an affine variety and a suitable G-module. The main tool with computing with invariants of linear algebraic groups is Derksen’s algorithm, which successively computes homogeneous invariants of increaseing degree, and reduces these to a minimal set via syzygy computations in modules over the underlying polynomial ring. – Invariant ring of an algebraic group defined by multivariate polynomials describing an affine variety – Construction of binary forms – Computation of invariants of given degree – Computation of fundamental invariants (Derksen algorithm)

13.5.5 Invariant Fields

If G is a group acting on a polynomial ring K[x1, . . . , xn], it also acts on the rational function G field K(x1, . . . , xn) by homomorphic extension. The invariants field K(x1, . . . , xn) is the field consisting of all functions which are fixed by G. Magma allows the construction of the invariant field and elementary computations within it. – Construction of invariant fields of finite groups or algebraic groups – Computation of fundamental invariants (Müller-Quade/Bethalgorithm)

13.5.6 Invariants of the Symmetric Group – Construction of elementary symmetric polynomials – Determination of whether a polynomial is symmetric – Presentation of a symmetric polynomial in terms of the elementary symmetric polynomials 76 Magma Overview V2.18 (December 16, 2011)

14 Algebraic Geometry

The algebraic geometry module includes both machinery for studying general algebraic varieties and facilities for working with more restricted varieties suach as algebraic curves and surfaces. The major categories include:

• Schemes and their maps • Coherent sheaves • Algebraic curves • Algebraic surfaces • Toric varieties • Graded rings and geometric databases

14.1 Schemes This module comprises general tools for working with schemes defined by polynomial equations in affine or projective space. Such tools include Gröbnerbasis computation, dimension and image of maps, and linear algebra calculations formalised as linear systems on projective space. Maps between spaces may also be constructed and studied.

14.1.1 Ambient Spaces A scheme is contained in some ambient space, either an aﬃne space or one of a small number of standard projective spaces. A characteristic of these spaces is that they have some kind of polynomial ring as their coordinate ring.

– Aﬃne and projective spaces including weighted projective space – Ruled surfaces – Rational scrolls – Direct products of ambient spaces – Points of schemes with coeﬃcients in k-algebras

14.1.2 Creation and Properties – Creation of schemes by equations or implicit methods – Saturation of the defining ideal of a scheme in the projective cases – Changing the base ring – Point set operations – Projective closure and affine patches – Global properties: Dimension, reducibility, singularity – Prime components, primary components – Local geometric analysis: singularities, tangent spaces – Zero-dimensional schemes – Determining whether a scheme over a number field is locally solvable – Searching for points (over the rationals). – Listing of all points oevr a finite field. Magma Overview V2.18 (December 16, 2011) 77

14.1.3 Mappings

– Construction of maps between spaces – Coordinate manipulation functions, including birational transformations of the projective plane – Calculation of pullbacks of schemes by maps – Tools to enable the calculation of images of schemes by maps – A large number of specialised types of schemes, especially curves, have additional map functionality – Availability of maps with multiple deﬁnitions – Alternative map type for ordinary projective schemes deﬁned by the “graph” of the map in the product scheme.

14.1.4 Automorphisms

Automorphisms of schemes defined over a field may be constructed. The main cases where there is significant functionality is for automorphisms of affine and projective spaces and algebraic curves.

– Construction of general automorphisms of affine space in terms of functions or matrices – Construction of special automorphisms of affine space: translation, permutation automorphisms, Nagata automorphism, projectivities – Construction of general automorphisms of projective space in terms of polynomials or matrices – Construction of special automorphisms of projective space: translation, quadratic transformation – Automorphism group of a projective space defined over a finite field – Automorphism groups of algebraic curves of genus at least 2 (see below).

14.1.5 Isomorphic Projections

– Computation of the tangent and secant varieties of schemes – Faster tests for whether a speciﬁed point lies in the tangent or the secant varieties of a scheme. – Isomorphic projection of projective schemes to smaller dimensional ambient spaces – Birational embedding of plane curves as non-singular projective space curves in 3-dimensional space

14.1.6 Linear Systems

The complete linear system on P of degree d is the collection of all homogeneous polynomials of degree d on P, or equivalently, the degree d hypersurfaces thereby deﬁned. A general linear system corresponds to some vector subspace of the coeﬃcient space of a complete linear system.

– Creation of a linear system explicitly – Creation of a linear system satisfying geometric conditions – Properties: Sections, degree, dimension, base scheme – Properties: Base component, base points, generic multiplicity at a point – Complement with respect to a subsystem or scheme – Intersection – Maps of a linear system 78 Magma Overview V2.18 (December 16, 2011)

14.2 Coherent Sheaves Machinery for operations related to coherent sheaves on projective schemes. Currently it is only available for ordinary projective schemes. There is functionality to obtain some important sheaves, like the canonical sheaf, and perform some basic constructions, like tensor powers or duals. The initial emphasis has been on invertible sheaves and their associated maps into projective space. The map of such a sheaf and its image may be computed. The invertible sheaf associated to an effective Cartier divisor D, given as a subscheme of the base scheme, can be computed and a Riemann-Roch space for D can be returned in explicit form. The dimension of cohomology groups of arbitrary sheaves can be computed. The maximal representing graded module of the sheaf - the full module of global sections of all twists - can also be computed, which is key to much of the functionality. There are various tests for properties like local-freeness. The package was developed in Sydney and the algorithms rely on a number of computational commutative algebra tricks well-known to the experts in the field. – Definition of a general sheaf via a graded module over the coordinate ring of the ambient scheme. – Computation of some special sheaves like the structure sheaf and the canonical sheaf of a locally Cohen-Macaulay equidimensional scheme. – Serre twists of a sheaf. – Computation of the maximal (saturated) module representing a sheaf. – Tensor products and powers, direct sums, duals and Homs. – Homomorphisms between sheaves on the same base scheme. Kernels, images and cokernels. – Test for local-freeness. – Test for a sheaf being arithmetically Cohen-Macaulay. – Test for and construction of isomorphisms between sheaves on the same base scheme. – The associated “divisor” map into projective space of an invertible sheaf and its image. Includes, for example, canonical, anticanonical and adjunction maps. – The Riemann-Roch space of an effective Cartier divisor D and an invertible sheaf isomorphic to O([D]). – Dimension over the base field of cohomology groups of twists of a sheaf, using the Decker-Eisenbud- Floystad-Schreyer algorithm based on the BGG correspondence. This can be used to get various standard invariants of a scheme. – Intersection pairing of the divisor classes represented by two invertible sheaves on a non-singular projective surface.

14.3 General Algebraic Curves Magma includes a general package for working with algebraic curves. These are schemes of dimension 1, a particular case being plane curves that are defined by the vanishing of a single polynomial in 2-dimensional affine or projective space. The main features of interest are for integral (reduced and irreducible) curves. For these Magma computes explicit representations of the curve’s field of rational functions allowing computations with places, divisors and differentials, calculation of the geometric genus, computation of Riemann-Roch spaces and more. This relies on the underlying function field machinery, but the results are available purely in the context of curve structures.

14.3.1 Construction and General Properties – Creation of affine and projective curves together with the ambient spaces – Basic manoeuvres between affine and projective curves: projective closure, affine patches and so on – Basic scheme-type functions: e.g., irreducibility – Specialised data types for distinct classes of curves such as elliptic curves – Linear systems of curves in the projective plane with assigned basepoints Magma Overview V2.18 (December 16, 2011) 79

– Implicitization of parametric curves – Parametrization of rational curves – Point searches on plane curves using a p-adic method of Elkies – Global invariants of curves, such as genus and dimension – Function field and divisor computations (see below) – Random curves by genus over the rationals or finite fields for all genera up to 13. – Index calculus for discrete logs on the Jacobians of plane curves.

14.3.2 Mappings

– Creation of the basic automorphisms of the affine plane (translation, flip, automorphism) and general maps between curves. – Evaluation of the image of a point under a rational map as a power series expansion – Ramification divisor of a non-constant dominant map – Pullbacks and pushforwards of functions, differentials or divisors along maps between curves. – Explicit computation of the map into ordinary projective space associated to a divisor (class) - in particular the canonical map. Optimised computation of canonical images.

14.3.3 Automorphism groups

– Computation of the full automorphism group (over the base field) of an (integral) algebraic curve of genus at least 2. – Structures for defining and working with groups of automorphisms of such curves. In particular, efficient arithmetic of automorphisms (products, powers, etc.) and subgroup definition. – Abstract realisation of an automorphism group as a permutation group with 2-way correspondence. This allows the full range of abstract group constructions (eg p-Sylow subgroups) to be applied. – Computation of the quotient of a curve by a group of automorphisms as a new curve with an explicit quotient scheme map. – Tests for isomorphism and construction of explicit isomorphisms as scheme maps.

14.3.4 Local Analysis

– Calculation of tangent spaces and cones for plane curves – Identiﬁcation of all singularities of a curve, together with the basic analysis – Blowups, including weighted blowups, of points on plane curves – Local intersections on plane curves

14.3.5 Ordinary Plane Curves

– Tests for only ordinary or only nodal singularities of plane curves. – Special computation of the canonical or more general adjoint maps for an ordinary plane curve. – Random nodal curves of specified degree and number of singularities. – Random ordinary curves of specified degree and with specified number of ordinary singularities of given multiplicities. – Special parametrization routines for ordinary rational plane curves and rational normal curves. 80 Magma Overview V2.18 (December 16, 2011)

14.3.6 Function Field The following functions apply to integral curves in general.

– Construction of the function field of a curve – Function field arithmetic – Exact constant field – Enumeration of places of degree m (over a finite field) – Residue class field – Class number – Divisor class group of a curve defined over a finite field – Group of global units of the function field of a curve

14.3.7 Divisors and the Riemann-Roch Theorem The following functions apply to integral curves in general.

– Creation of places and divisors and their arithmetic – Compute the divisor of a function on a curve and other constructors – Determine whether a divisor is principal and if so ﬁnd a function with the given divisor – Ramiﬁcation divisor – Riemann–Roch space of a divisor – Computation of Weierstrass places on a curve

14.3.8 Diﬀerentials The following functions apply to integral curves in general.

– Arithmetic and various, related operations – Valuation of a differential at a place – Divisor of a differential – Differential spaces for given divisors (e.g. holomorphic differentials) – Higher differentiations – Residue of a differential at a place of degree one – Cartier operator and representation matrix of the Cartier operator (curves over finite fields)

14.3.9 Resolution Graphs and Splice Diagrams – Creation of resolution graphs and their vertices, either implicitly using resolution routines or Newton Polygons, or explicitly by listing the required data – Calculation of numerical data associated to blowups, e.g., the canonical class of certain rational surfaces – The Cartan matrix of a graph and associated calculations such as the contribution to the genus of a plane curve of a singularity having a given graph as its resolution – Surgery on resolution graphs such as cutting an edge of a graph – Creation of splice diagrams implicitly and explicitly – Edge determinants and linking numbers of splice diagrams – Test for regularity of splice diagrams – Translation between resolution graphs and splice diagrams Magma Overview V2.18 (December 16, 2011) 81

These decorated graphs encode data generated by the resolution machinery. At present they are only attached to resolutions of plane curve singularities, but in due course they may be extended to resolutions of linear systems, surface singularities, special ﬁbres in curve ﬁbrations or other geometrical contexts. The package includes a resolution function for curves adapted to present its output in resolution graph format. This does not supersede other resolution machinery such as Puiseux expansions or genus calculations, but is intended as a complementary tool for those users with this kind of geometric background.

14.4 Algebraic Surfaces Magma now contains special functionality for general projective hypersurfaces in P3 and other speciﬁc types of algebraic surface like anticanonically-embedded Del Pezzo surfaces. The general hypersurface package was developed in Linz by Josef Schicho and Tobias Beck. It’s main aim is the formal desingularisation of 2-dimensional hypersurfaces. This allows the computation of invariants like the geometric plurigenera and more generally gives explicit adjoint linear systems that represent the canonical, anticanonical or adjunction mappings amongst others. The package also contains a function to classify rational hypersurfaces and map them to terminal types. This combines with the other main component of the current surface functionality, which is speciﬁc code to parametrize terminal rational surfaces. The major part of this is packages for parametrizing Del Pezzo surfaces, which were developed in joint work by Schicho’s group and the Magma group and use the Lie algebra method for the high degree Del Pezzos. The sheaf module also provides some useful general functionality that may be applied to locally Cohen-Macaulay schemes in ordinary projective space representing surfaces (see above).

14.4.1 Formal Desingularisation and Classification – Formal resolution of singularities of hypersurfaces in 3-D projective space over a field of characteristic zero, following the Hirzebruch-Jung method. – Result given using algebraic power series for the formal completions of components over the singular locus. – Geometric genus, arithmetic genus and higher plurigenera of (any) desingularisation. n – General KX (m) adjoint linear systems. – Tests for a hypersurface being rational or (birationally) ruled. – Classification of rational surfaces by terminal type. Construction of birational map to terminal model following Schicho’s algorithm.

14.4.2 Parametrization Of Rational Surfaces – Test for the parametrizability over the rational numbers of a rational hypersurface in 3-D projective space and construction of an explicit parametrization. – Routines for the direct parametrization of singular or non-singular anticanonically embedded Del Pezzo surfaces of degree at least 5 over the rationals. Uses the Lie algebra method for degree greater than 5. – General routine for direct parametrization of any degree Del Pezzo (singular or non-singular) in its standard weighted anticanonical ample embedding by blowing down lines to reduce to degree at least 5. – Routines for the direct parametrization of quadric hypersurfaces, rational pencils and conic bundles. 82 Magma Overview V2.18 (December 16, 2011)

14.5 Toric Varieties There are many points of view on toric geometry and many diﬀerent uses. The applications moti- vating this implementation come from low dimensional birational geometry and mirror symmetry; it is capable of doing substantial algebraic geometry, as well as the usual array of combinatorial routines associated to polytopes, cones and fans. In particular the construction of any general toric variety is possible; you are not limited to the standard subclasses such as smooth or Q-factorial examples. Toric varieties can also be used as ambient spaces for other more complicated varieties (in the same way as aﬃne and projective spaces).

14.5.1 Toric lattices One often begins to study toric geometry by considering a lattice L = Zn and discussing polygons or cones in its overlying rational vector space LQ = L ⊗ Q. In Magma we bind these two spaces together in a single object, a toric lattice. Although these spaces lie underneath all the combinatorics, it is seldom necessary to be explicit about them; they often remain invisible. Magma supports many operations for constructing and manipulating toric lattices, including: – Creation of a toric lattice and its dual; – Construction of direct sums, sublattice, and (torsion free) quotient lattices; – The sublattice ﬁxed under a group action; – Maps between toric lattices; – Standard operations on points in a toric lattice.

14.5.2 Cones and Polyhedra A cone (in a toric lattice L) is the convex hull of finitely many rays. A (rational) polytope is the convex hull of finitely many points of L. More generally a (rational) polyhedron is given by the Minkowski sum of a polytope and a cone. There exist dual definitions in terms of the intersection of finitely many half-spaces. There is no requirement that polyhedra are of maximum dimension in the ambient toric lattice. Important operations include: – Standard constructions of cones, half-spaces, cross polytopes, cyclic polytopes, etc.; – Creation of cones and polyhedrons via a system of inequalities; – The defining inequalities of a cone or polyhedron; – The dual cone or polyhedron; – Taking the (affine) linear space spanned by, or contained in, a polyhedron; – Standard properties such as dimension, volume, and boundary volume; – Triangulations of a polytope and of its boundary; – Information on whether a polyhedron is simplicial, reflexive, terminal, etc.; – Construction of the cone in L × Z spanned by a polyhedron P in L; – The polyhedron defined by taking the convex hull of P ∩ Zn for any polyhedron P ; – The infinite and compact parts of a polyhedron; – Calculation of intersections, Minkowski sums, translations, etc.; – Taking hyperplane slices; – The minimum Q-generators and Z-generators of a cone; – Point membership; – Point counting and enumeration of lattice points; – The Ehrhart series, Ehrhart polynomial, and δ-vector of a polytope; – Faces of a cone or polyhedron, the f-vector and h-vector; – Construction of the face-poset; – The group of lattice automorphisms fixing a polytope. Magma Overview V2.18 (December 16, 2011) 83

14.5.3 Fans A fan F (in a toric lattice L) is a collection of cones in L satisfying the usual conditions of a cell decomposition. Magma allows for the construction of any fan; it will check that the cones intersect correctly, and will add the lower-dimensional cones as necessary. There are also several constructors for well-known fans, and standard methods for modifying existing fans. – Creation of a fan from a collection of cones; – Constructions for the fan of aﬃne space, projective space, and weighted projective space; – Construction of the product of two (or more) fans; – The dual fan and spanning fan generated by a polyhedron; – The normal fan to a cone C, given by the quotient of L by the span of C; – Support and point membership; – Extraction of the rays, one-skeleton, singular cones, maximum dimensional cones, and so forth; – The blowup of F at a point in L; – Answers to questions such as whether F is projective, or nonsingular, or Q-factorial; – The simplicial subdivision of a fan; – A reﬁnement of F such that the resulting fan is nonsingular; – Maps between fans.

14.5.4 Basic operations on toric varieties There exist many standard constructors for toric varieties. They can also be defined directly from a fan, or in terms of a Cox ring. However one constructs a toric variety, the corresponding fan can always be recovered, although this is seldom necessary. Once you have constructed your toric variety you can treat it as a purely geometric object, as you would any scheme in Magma. Operations include: – Standard constructions for affine, projective, and weighted projective space; – Construction of scrolls, or of any toric variety specified by a collection of gradings; – The toric variety given by the product X × Y ; – Recovery of a fan from any toric variety, and of the toric variety associated to any fan; – The variety corresponding to a Cox ring can be constructed, and the Cox ring of any toric variety recovered; – Extract the i-th affine patch of a toric variety X, along with the inclusion map; – A resolution, Q-factorialisation, terminalisation, etc., along with the corresponding maps; – Extract the one parameter subgroup lattice, monomial lattice, Cox monomial lattice, or divisor class lattice; – Is the variety complete? Is it projective? Is it Fano? – Determine whether the toric variety is nonsingular, Gorenstein, Q-factorial, terminal, or canonical.

14.5.5 Maps of toric varieties Magma supports the creation of very general maps between toric varieties. These can be derived from maps between the associated toric lattices, or presented in terms of the variables of the Cox ring. More generally one can deﬁne all maps between toric varieties using an appropriate notion of ‘rational radical function’ deﬁned in terms of the polynomial Cox coordinates. Common questions Magma can answer include: – Are two toric varieties naturally isomorphic? – Is a given map regular? – The indeterminancy locus of a map. 84 Magma Overview V2.18 (December 16, 2011)

14.5.6 Invariant divisors and Riemann-Roch spaces Divisors on toric varieties work in the same way as on other varieties, except that within each linear equivalence class it is possible to choose torus invariant representatives. Divisors on a toric variety have a single divisor group as their parent; divisors can be constructed by coercing the appropriate data into this group, or via several standard constructions. Once you have constructed your divisor, Magma supports all the operations you would expect. Examples are: – Construction of the zero divisor and canonical divisor; – Construction of a “nice” representative of an element in the Picard latice or divisor class lattice, chosen to be as eﬀective as possible; – The combinatorial data deﬁning a Weil or Cartier divisor; – Is a given divisor principal or Q-principal? Is it linearly equivalent to a Cartier divisor? – Are two divisors linearly equivalent? – Answers to questions such as whether a divisor D is Cartier, Q-Cartier, Weil, ample, very ample, nef, or big; – The Picard class or movable part of a divisor D; – Proj of the ring of sections of D; – The relative (sheaf) Proj of sections of the divisor D; – The Riemann-Roch polytope associated to a divisor; – The basis of the associated Riemann-Roch space; – The graded cone of sections of multiples of D; – The Hilbert series and Hilbert polynomial of (X,D).

14.5.7 Mori theory The Mori cone and nef cone of a toric variety can be computed, extremal rays calculated, and so on. Magma can answer questions such as whether a given extremal ray corresponds to a Mori ﬁbre space, is a divisorial contraction, or gives a small contraction. Standard constructions and operations are also supported: – The images of extremal contractions of rays in the nef cone; – A description of the extremal contractions; – The (generalised) ﬂip of the morphism given by the Q-Cartier divisor;

– The weights of the Gm action whose variation would give the ﬂip; – An implementation of the Minimal Model Programme. Magma Overview V2.18 (December 16, 2011) 85

14.6 Graded Rings and Geometric Databases The computation of generic families of K3 surfaces embedded in weighted projective spaces of small dimension is a project that has been running since the mid-1970s. Until two years ago, a main result was a collection of about 400 families of K3 surfaces (which had been recorded as a database in Magma). Magma now contains a database of 24,099 K3 surfaces (including correcting half a dozen mistakes in codimension 4) that, in a well-defined sense, contains a sketch of all possible examples. One main point is that these results are a major step on the way to understanding Fano 3-folds and their birational automorphisms. These are the 3-dimensional analogues of rational curves or Del Pezzo surfaces and are a very exciting research frontier. It is possible that the final classification of Fano 3-folds (expected to contain a few thousand families) will almost be contained in the K3 database. The methods applied to K3 surfaces work in many other contexts. Currently Magma contains the calculations for: subcanonical curves, K3 surfaces, Fano 3-folds and Calabi–Yau 3-folds; other cases are in progress. The basic method is to assemble a lot of data that should occur on these varieties, feed it into the Riemann–Roch formula to get a Hilbert series, and then attempting to describe a plausible variety embedded in weighted projective space that has that Hilbert series. The ideas are described in the paper [1], while improved handling of high codimension possibilities using projection/unprojection theorems is described in [8]. Other cases are covered by the papers [11], [37]. Notice that these routines do not describe explicit graded rings: they do not explain how to write the equations, but only say which weighted variables should be used in the equations. (The Hilbert series does include more information, but still much less than would dictate explicit equations, in general.) The process of recovering equations through projections is called ‘unprojection’, and that is still some way off being implemented as a method in commutative algebra. These families exhibit large Gorenstein rings with as yet unknown structure. The search for structure theorems for Gorenstein rings occupied much of the 1970s after the Buchsbaum–Eisenbud structure theorem in codimension 3. However no substantial new cases were discovered. These examples, not available in the 1970s, are a major motivation for a renewed assault on this problem. – Functions that create subcanonical curves, K3 surfaces, Fano 3-folds and Calabi–Yau 3-folds from data associated to the Riemann–Roch formula in each case – A K3 database containing 24,099 candidates for polarised K3 surfaces. There are also smaller databases of Fano 3-folds as a first step in the final classification – Functions for interrogating databases, including functions that can add new entries or rewrite existing entries 86 Magma Overview V2.18 (December 16, 2011)

15 Arithmetic Geometry

The area of number theory known as “arithmetic algebraic geometry” is mainly concerned with studying algebraic varieties over a non-algebraically closed base field, especially properties that are highly sensitive to the base field. For instance, one seeks to understand the set of rational points and other rationality questions, points over local or finite fields, and associated L-functions. Magma has extensive packages dealing with arithmetic features for several much-studied classes of curves. Many algorithms of interest for curves over global fields are implemented not only over Q but also over number fields or function fields. The main packages are:

• Rational curves and conics • Elliptic curves • Genus one curves • Hyperelliptic curves • L-functions

15.1 Rational Curves and Conics Rational curves and conics in Magma are nonsingular plane curves of degree 1 and 2, respectively. The central functionality for conics concerns the existence of points over the rationals. If a point is known to exist, then a conic can be parameterized by a projective line or a rational curve. A rational curve in Magma is a linearly embedded image of the projective line, to which the full machinery of algebraic plane curves may be applied. Tools provided include the local-global theory, the existence of points on conics, efficient algorithms for finding rational points, parametrizations and isomorphisms of genus zero curves. – Computation of a conic model, for an arbitrary curve of genus zero – Diagonalization of a given conic over a general field – Over number fields, a careful diagonalization routine which avoids making the discriminant harder to factor – Routines to decide whether a conic is solvable, over Q, number fields and function fields; calculation of Hilbert symbols – Solution of conics over Q using Simon’s algorithm (this involves LLL-reduction of an indefinite form, and includes Cremona-Rusin reduction as a special case) – Solution of conics over rational function fields using the algorithm of Cremona-van Hoeij – Solution of conics over number fields using an algorithm based on Lagrange’s method plus other techniques – Reduction of a rational point on a conic so that its coordinates satisfy Holzer’s bounds (using an algorithm due to Mordell) – Parametrization of a general conic using a known point – Simon’s parametrization for conics over Q Magma Overview V2.18 (December 16, 2011) 87

15.2 Elliptic Curves Magma contains an extensive module for computing with elliptic curves. The reader should note that elliptic curves inherit from the general plane curve datatype so that all plane curve operations are applicable in addition to those listed below.

15.2.1 Construction and Properties – Creation of an elliptic curve over a field – Creation of an elliptic curve with given j-invariant – Creation of an elliptic curve from a general genus 1 curve – Models: Weierstrass form, integral model, minimal model, simplified model – Arithmetic with rational points, including division – Extension and lifting of curves induced by maps of base rings – Function fields, divisors, places – Invariants: b-invariants, c-invariants, j-invariant, discriminant – Division polynomials – Subgroups and subschemes of elliptic curves as separate types

15.2.2 Morphisms – Extension and lifting of maps induced by change of base ring of curves – Isomorphisms, isogenies and rational maps between curves, translation maps on a curve – Isogeny operations: degree, composition, construction with given kernel, Frobenius endomorphism – Kernel and image of an isogeny as subgroups, image and preimage of a subgroup under an isogeny – Endomorphisms – Isomorphism operations: inverses, composition – Testing for isomorphic curves over fields with root-finding – Testing for isogenous curves over finite fields via point-counting, and over the rationals via kernels (a map is returned also here)

15.2.3 Features over finite fields Magma contains implementations of various methods for point-counting, each optimized for fields in a certain range. In addition, there are efficient implementations of some items of cryptographic interest, including pairings and GHS. – Characterization of ordinary and supersingular elliptic curves – Representative supersingular curve – Enumeration of all points (small fields), random point – Quadratic twist, all quadratic twist, all twists – Order of a point via baby-step–giant-step for small p – Schoof-Elkies-Atkin (SEA) algorithm for finding the order of the group of rational points – Schoof-Elkies-Atkin (SEA) algorithm with early abort facility for searching for cryptographically secure curves – Lercier’s extension of the SEA algorithm – p-adic canonical lift methods for determining the number of points over finite fields of small characteristic p (p up to around 50). – p-adic deformation methods for determining the number of points over finite fields of smallish characteristic p not covered by canonical lift. 88 Magma Overview V2.18 (December 16, 2011)

– Structure of the abelian group of rational points – The zeta function of a curve – Discrete logarithm (parallel collision search version of the Pollard rho algorithm) – Weil, Tate and Eta pairings – Gaudry-Hess-Smart (GHS) descent with explicit map from the curve to divisors (or to the Jacobian of a hyperelliptic curve).

For a random curve taken over a 168-bit prime ﬁeld GF (p), Magma takes an average of 47 seconds to determine the order of the group. In the case of a random curve taken over a 400-bit prime ﬁeld the average runtime is 2200 seconds. In characteristic two, Magma takes around 0.03 seconds for a curve over GF (2162), where Gaussian Normal Bases are available and 0.12 seconds over GF (2160). Over GF (2268) (again with GNB) it takes around 0.06 seconds.

15.2.4 Features over Q – Invariants: conductor, regulator, periods, and period lattice – Torsion subgroup – Tate’s algorithm for computing local information, Kodaira symbols and Tamagawa numbers – Twist of an elliptic curve by an integer, minimal twist – Theory of heights: local height, naive height, canonical height, height pairing – Bounds on the “height difference”: Silverman’s bound and Siksek’s bound – p-adic heights – Elliptic logarithm, p-adic elliptic logarithm – Elliptic exponential (℘-function) using Newton iteration – L-series, and special values of the L-series, – Analytic rank – Root numbers – modular degree – Isogeny class, computation of isogenies – Heegner points for rank 1 curves, and general modular parametrisation – Rigorous determination of the finite set of S-integral points, for a given finite set of prime S. As an application, determination of the S-integral points on other genus 1 curves (quartic, Ljunggren and Desboves curves). – John Cremona’s database of all elliptic curves over Q having conductor up to 130 000

15.2.5 Rational points over Q Magma contains an unrivalled array of techniques that are used for determining the Mordell-Weil group of rational points E(Q) for a curve over Q (including the analytic tools already listed). These are applied in various combinations to obtain bounds on the Mordell-Weil rank, and to find generators. – 2-descent using the “invariant method” (the algorithm that is implemented in John Cremona’s program “mwrank”) – 2-descent using the “algebraic method” (recently re-implemented efficiently) – Cassels-Tate pairing between 2-coverings (fast algorithm of Donnelly) – 4-descent above a given 2-covering (algorithm of Merriman-Siksek-Smart, with refinements) – Cassels-Tate pairing to resolve 4-coverings (algorithm of Donnelly) – 8-descent above a given 4-covering (algorithm of Stamminger and Fisher) Magma Overview V2.18 (December 16, 2011) 89

– 3-descent (algorithms of Cremona-Fisher-O’Neil-Simon-Stoll and Schaefer-Stoll) – 6- and 12-descent as combination of 2-, 3- and 4-descent (Fisher) – Descent using 2-isogenies or 3-isogenies. – Minimisation and reduction of the covering curves in each of these cases (algorithms mainly due to Cremona, Fisher and Stoll) – Searching on for rational points on the elliptic curve, and on 2-coverings, using sieving – Point-searching on 4-coverings using a dedicated implementation of Elkies p-adic method – Point-searching on other coverings using the general-purpose implementation of Elkies p-adic method

15.2.6 Features over number ﬁelds – Tate’s algorithm, local information and local minimal model – Conductor – Global minimal model in the case where one exists – Torsion subgroup (and torsion bounds, p-torsion subgroup) – Height machinery – Analytic rank – L-functions – Determining whether curve has complex multiplication (this returns the discriminant of the CM order). – “Elliptic curve Chabauty” methods, with Mordell-Weil sieving (algorithm of Bruin)

15.2.7 Rational points over number ﬁelds Some of the techniques for Mordell-Weil groups over Q are also implemented for number ﬁelds. – 2-Selmer groups, 2-descent – 2-isogeny Selmer groups – Cassels-Tate pairing between 2-coverings (algorithm of Donnelly) – 4-descent, using code for “descent on a 2-covering” (Bruin, Stoll) – Point-searching on elliptic curves and 2-coverings

15.2.8 Features over function fields Many of the same features are also implemented for elliptic curves over univariate function fields and their finite extensions. (However, many of the individual routines impose further restrictions: for instance, only allowing fields with finite base field, or odd characteristic, or only rational function fields.) – Tate’s algorithm, local information and local minimal model – Conductor, global minimal model – Torsion subgroup (and torsion bounds, p-torsion subgroup) – Height machinery – L-functions – Analytic rank bounds

15.2.9 Rational points over function ﬁelds – 2-descent in odd characteristic – Cassels-Tate pairing between 2 coverings in odd characteristic – 2-descent (via isogenies) for ordinary curves in characteristic 2 – Computation of the Mordell-Weil group via the Neron-Severi group of the elliptic surface in cases where this is a rational surface. 90 Magma Overview V2.18 (December 16, 2011)

15.3 Models of Genus One Curves This package deals “genus one normal models” of degree n = 2 (hyperelliptic curve), n = 3 (plane cubic), n = 4 (quadric intersection in P 3) or n = 5 (in which case the model is given in a more complicated way). An element of order n = 2, 3, 4 or 5 in the Tate-Shafarevich group of an elliptic curve can be expressed as a curve in this form. The package provides a variety of routines based on the invariant theory of these objects, as well as routines for obtaining “nice” models (for models over Q) by minimisation and reduction. Most of the algorithms are due to Fisher, Cremona and Stoll. – Creation of genus one models of degree 2, 3, 4 or 5 from various input – Extraction of deﬁning data, equations for the curve – Standard a, b and c “invariants” of a model; the Jacobian and n-covering map – Polynomial formulas given for families of elliptic curve with the same n-torsion as a given curve. – Handling of transformations of genus one models – Minimisation (over Q and rational function ﬁelds) – Reduction (over Q for n = 2, 3 and 4) – Addition of models, regarded as elements of the Weil-Chatelet group – Testing equivalence of two given models over Q – Local solubility testing for models over Q Magma Overview V2.18 (December 16, 2011) 91

15.4 Hyperelliptic Curves and Jacobians Hyperelliptic curves in Magma are a special type of curve; they are given as nonsingular curves in a naturally weighted projective plane. They inherit all operations that are valid for general curves. In contrast, Jacobians are not implemented as a type of scheme or variety; instead they are implemented in terms of divisors on the curve. In particular, Jacobians are not defined by a set of algebraic equations, or by an embedding. They do not inherit from the general scheme datatype. The broad aims of the package are as follows. One wants fast arithmetic on Jacobians (operations with normalized divisors) over general fields. Over finite fields, one wants methods for determining the number of points on curves and on their Jacobians, and more generally the zeta functions. Over global fields, one is mostly interested in the group of rational points on a Jacobian and the (finite) set of rational points on a curve, as well as local properties, heights and so on. The first version of this package was developed by Michael Stoll; others who made major contributions include P. Gaudry, E. Howe, and several past and current members of the Magma group.

15.4.1 Construction and Properties

– Standard quadratic models, simplified models – Creation of isomorphisms, automorphisms, and operations with these – Test for curves being isomorphic – Function fields, divisors, places – Invariants: degree, genus, discriminant – Igusa and related invariants – Construction of a curve given its Igusa invariants (over suitable fields) – Extension and lifting of curves induced by maps of base rings – Determination of points at infinity – Determination of whether a general algebraic curve C is hyperelliptic and, if so, computation of a Weierstrass model for C – Determination of whether a general algebraic curve C is geometrically hyperelliptic - ie, is hyperelliptic over the algebraic closure of the ground field.

15.4.2 Operations on curves over ﬁnite ﬁelds

– Enumeration of all rational points (for small ﬁelds) – Random point – Quadratic twist, all quadratic twists – Counting points – Zeta function

15.4.3 Operations on curves over Q and number ﬁelds

– Integral model, minimal Weierstrass model, p-normal model, reduced model – Reduction types and conductors of a genus 2 curve at odd primes – Local solubility testing (optimized code for the case of hyperelliptic curves) – 2-descent “on the curve” giving a bound on the set of rational points, often better than 2-descent on the Jacobian (algorithm of Bruin and Stoll) 92 Magma Overview V2.18 (December 16, 2011)

15.4.4 Construction of Jacobians – Creation of the Jacobian of a given curve as the divisor class group – Creation of points – Normal form and addition of points – Fast addition algorithm of R. Harley for genus 2 curves – Analytic model of the Jacobian: translation between the algebraic and analytic Jacobians

15.4.5 Kummer Surface of a genus 2 Jacobian The Kummer surface is a certain quotient of a genus 2 Jacobian. In Magma, it is part of the data associated to the Jacobian. However, unlike the Jacobian itself, the Kummer surface is given by a defining equation. – The defining equation of the surface – Creation of points – Pseudo-addition, and duplication of points – Preimage on the Jacobian of a given point on the Kummer surface – Search for rational points having three coordinates specified

15.4.6 Operations on Jacobians over ﬁnite ﬁelds The p-adic techniques used for point counting in small characteristic p are Kedlaya’s algorithm (p odd) and Mestre’s canonical lift method as adapted by Lercier and Lubicz (p = 2) [implemented by M. Harrison]. The other techniques for point counting are due to P. Gaudry and R. Harley and have been implemented in Magma by P. Gaudry.

– Enumeration of all rational points (small fields), random point – Order of a point via the Shanks or the Pollard-rho algorithms – Counting points: p-adic methods of Kedlaya and Mestre/Lercier/Lubicz – Counting points: p-adic deformation method. – Counting points: Shanks and Pollard methods – Counting points: An index calculus method for when the genus is large compared to the base field – Counting points: Schoof algorithm for finding the order modulo small primes in the case of a genus 2 curve – Counting points: Class group methods using the function field machinery. – Structure of the abelian group of rational points – Weil pairing of points

15.4.7 Operations on Jacobians over Q and number ﬁelds Some of the functions listed here apply only for Jacobians of genus 2 curves, while others also apply for higher genus hyperelliptic curves. – Order of a point – 2-torsion subgroup, full torsion subgroup – Naive height and canonical height of a point, bounds on the “height diﬀerence”, the height pairing matrix and regulator – Enumeration of all Q-rational points up to a given height bound (Elkies-Stahlke-Stoll method) – Rank of the subgroup of the Mordell-Weil group generated by a given set of points on the Jacobian – Computation of the 2-Selmer group, giving a bound on the Mordell-Weil rank Magma Overview V2.18 (December 16, 2011) 93

– Chabauty’s method combined with Mordell-Weil sieving allows determination of the set of rational points on a genus 2 curve over Q with Mordell–Weil rank 1, assuming the Mordell-Weil group of the Jacobian is known. – Analytic representation of a Jacobian; conversion of points between the representations; computation of analytic invariants Some of this functionality is also implemented over general number ﬁelds, including the torsion routines, the height machinery, and the 2-Selmer group.

15.5 L-Series

There are many arithmetic objects that can have L-functions attached to them, and the ability to compute with these then opens up a wide array of computational possibilities. For instance, one can study special values, or look at the distribution of zeros. The Magma package for L-functions is largely due to Tim Dokchitser, and contains a variety of nontrivial functionality for many objects. There is also the ability for the user to self-build L-functions, either for individual objects, or for classes of them. The main prerequisite for this is the ability to compute Euler factors. The classical ζ-functions of Riemann and Dedekind, and their generalisations by Dirichlet and Hecke, form the first examples (these are GL(1) in some sense), with elliptic curves and modular forms giving examples on GL(2). Magma can also compute with these over number fields other than the rationals, with Artin representations being available (these are useful when the Galois closure is small), possibly through tensor products. The ability to take some tensor products and symmetric powers also exists, though the computations at bad primes are not yet fully automated, and the generality still needs to be expanded. Many examples are given in the handbook. – Riemann ζ-function – Dirichlet L-functions – The Dedekind ζ-function of a number field – Hecke L-functions (including for Grössencharacters) – L-functions of Artin representations – Hasse-Weil L-functions (associated to elliptic curves) – L-functions associated to modular forms (newforms) – “Tensor products” of L-functions – Symmetric power L-functions – Facility to construct user-defined L-functions (with coefficients determined by a given function) 94 Magma Overview V2.18 (December 16, 2011)

16 Modular Arithmetic Geometry

• Modular curves • Congruence subgroups of PSL(2, R) • Arithmetic Fuchsian groups • Modular forms • Modular symbols • Brandt modules • Supersingular divisors on modular curves • Modular Abelian varieties • Hilbert modular forms • Modular forms over imaginery quadratic ﬁelds

• Admissible representations of GL2(Qp)

16.1 Modular Curves A modular curve in Magma is an affine curve given by an equation in a standard form. These equations are stored in precomputed databases. The modular curves currently available are defined by “modular equations” of three kinds, each of which is a bivariate polynomial relation between the j-invariant and another standard function on X0(N). These give singular plane affine models for X0(N). One application for which these models are suited is to compute isogenies of degree N between elliptic curves. Features: – Creation of a modular curve of specified level from a database. Possible model types are Atkin, Canonical and Classical. – Database of modular equations: Atkin, Canonical and Classical.

– Parametrization of the isogenies of an elliptic curve by points on some X0(N). – The j-invariant of a modular curve as a function on the curve. – Automorphisms: Atkin–Lehner involutions. – Hilbert and Weber class polynomials.

16.2 Congruence Subgroups of PSL(2,R) + The group GL2 (R) of 2 by 2 matrices deﬁned over R with positive determinant acts on the upper half complex plane H = {x ∈ C|Im(x) > 0}} by fractional linear transformation: µ ¶ a b az + b : z 7→ . c d cz + d

+ Any subgroup Γ of GL2 (R) also acts on H.A fundamental domain for the action of a (discrete) subgroup Γ is a region of H∗ containing a representative of each orbit of the action. Magma contains a package written by Helena Verrill for working with H∗ and with congruence subgroups ∗ and their action on H . The congruence subgroups currently supported are Γ0(N), Γ1(N), Γ(N), Γ1(N), Γ0(N), and intersections of these. The main routines are for computing generators and fundamental domains for these groups, as well as data such as coset representatives.

– Calculations on the upper half complex plane H: distances, angles and geodesics

– Action of PSL2(R) on H: ﬁxed points and stabilizers – Construction of the congruence subgroups listed above Magma Overview V2.18 (December 16, 2011) 95

– Construction of cusps, cusp widths, and elliptic points of congruence subgroups – Computation of a fundamental domain for the action of a congruence subgroup, described by the vertices, and Farey symbols – Equivalence of points under the action of a congruence subgroup – Computation of generators of congruence subgroups, and coset representatives – Graphics: postscript output of pictures of fundamental domains, points and geodesics, and polygons with geodesic edges (all on the upper half complex plane)

16.3 Modular Forms Since V2.8, Magma has included packages for modular forms and modular symbols. These were originally developed by William Stein, and are continually being developed further and improved by the Magma group. The modular forms package is, to a large extent, built on top of the modular symbols package. However, it also contains several independent features, notably Eisenstein series, half-integral weight forms and weight 1 forms.

– Construction of spaces of modular forms of weight k ≥ 1/2 on Γ0(N) or Γ1(N) (or with specified character) – Decomposition into Eisenstein, cuspidal, and new subspaces – Computation of dimensions (by formulae) – Computation of bases of these spaces, expressing basis elements as q-expansions with desired number of terms – Arithmetic operations for modular forms – Computation of Hecke operators and Atkin-Lehner operators – Decomposition into invariant subspaces with respect to these operators – Characteristic polynomials of Hecke operators – Determination of all newforms of given level (with Fourier coefficients given in suitable number fields) – Determination of all reductions of newforms modulo a given prime

16.4 Modular Symbols Modular symbols provide explicit representations of homology groups associated to modular curves, which are suited to eﬃcient computation. Computing the Hecke action on these groups yields q-expansions of modular forms. Other computations yield arithmetic information about the Jacobians of modular curves and their irreducible factors. This package contains many routines of this kind, which are used by the other packages, and which may also be called directly. (There is some duplication, and in some instances, the routines in this package are “lower level”.) Many of the algorithms implemented here are described in William Stein’s thesis [46] or in [47]. Additional references are given in the Magma Handbook.

– Construction of spaces of modular symbols of given character, level, and weight – Computation of Hecke operators and Atkin-Lehner operators – Decomposition into invariant subspaces under these operators – Computation of degeneracy maps, and standard subspaces – Determination of twists of minimal level, for an eigenform – Computation of certain invariants of modular abelian varieties – The intersection pairing on the integral homology of modular curves – Special values of L-functions and computation of complex period lattices 96 Magma Overview V2.18 (December 16, 2011)

16.5 Modular Abelian Varieties

Modular abelian varieties over Q are the Q-irreducible quotients of Jacobians of modular curves X0(N) or X1(N). In Magma, modular abelian varieties are viewed as explicit quotients or subvarieties of these modular Jacobians. The implementation is built on the modular symbols package. (The algorithms do not involve explicit equations for the Jacobians as varieties, which would be impractical).

– Construction of modular abelian varieties associated to Hecke-invariant subspaces of modular forms. – Finite direct sums and quotients may be formed. – Explicit computation of the group Hom(A, B) or the ring End(A), as a subgroup of homology, for modular abelian varieties A, B over Q. – Computation of kernels, cokernels, and images of homomorphisms of abelian varieties. – Intersections of subvarieties. – Computation of discriminants of subgroups of endomorphism rings, such as Hecke algebras. – Upper and lower bounds on the order of the K-rational torsion subgroup of A. – The determination of whether or not two modular abelian varieties are isomorphic (in some cases). – Characteristic polynomial of Frobenius. – Tamagawa numbers and component group orders (in some cases). – Computation with torsion points as elements of rational homology. – Computation of all inner and CM twists (not provably correct). – Computation of “building blocks”.

16.6 Brandt Modules Brandt modules provide an alternative approach to computing modular forms: they admit a natural Hecke action, and can be identified with spaces of classical modular forms having the same Hecke action. Brandt modules are defined in terms of ideals of Eichler orders in quaternion algebras. Note that this package is only for Brandt modules over Q. A separate implementation for arbitrary totally real fields, using a more efficient algorithm, is at the core of the package for Hilbert modular forms (described below). Features:

– Construction of the Brandt module on the left ideal class of an Eichler order in a deﬁnite quaternion algebra over Q. – Elementary invariants: level, discriminant, conductor, etc. – Calculation of dimensions by standard formulae. – Arithmetic operations for module elements. – Construction of Hecke and Atkin Lehner operators on Brandt modules. – Decomposition of a Brandt module into invariant subspaces. – The Eisenstein subspace and the cuspidal subspace of a Brandt module. – Operations on subspaces: Orthogonal complement, intersection. – Properties of subspaces: Eisenstein, cuspidal, decomposable. – Inner product of elements with respect to the canonical pairing on their parent. – q-expansions associated with a pair of elements of a Brandt module. Magma Overview V2.18 (December 16, 2011) 97

16.7 Supersingular Divisors on Modular Curves Another construction of interesting Hecke-modules is via the Hecke action on divisors on the supersingular points on X0(N) in characteristic p. More precisely, the module considered is the free abelian group on the supersingular elliptic curves in characteristic p enhanced with level N structure. It is computed using the “method of graphs” of Mestre and Oesterl´eand the Brandt modules algorithm. – Computation of Hecke operators and Atkin-Lehner involutions on modules of supersingular divisors. – Decomposition of a Brandt module into invariant subspaces with respect to these operators. – The monodromy pairing.

16.8 Hilbert Modular Forms Since V2.15, Magma has included a package for computing Hecke operators on spaces of Hilbert modular forms, over arbitrary totally real fields and for arbitrary level. Two separate algorithms are implemented (both rely on the Jacquet-Langlands correspondence, but they make use of different kinds of quaternion algebras). The algorithm due to Dembele is available for all weights greater than or equal to 2; it is an efficient approach to Brandt module computations. The implementation of the core calculations is fast, and this method has been used for fields of degree up to 10. The algorithm due to Greenberg and Voight is available for parallel weight 2, and makes use of Voight’s algorithm for fundamental domains of Fuchsian groups associated to Shimura curves. – Construction of cuspidal spaces of Hilbert modular forms over any totally real field, of given level and weight – The quaternion order which underlies the computations for a particular space may also be specified – Computation of dimensions (by “formulae”, or by computing the space) – Computation of Hecke operators – Computation of new subspaces – Decomposition into invariant subspaces with respect to the Hecke action

16.9 Modular Forms on Imaginary Quadratic Fields Since V2.16, Magma has included a package for modular forms of weight 2 over arbitrary imaginary quadratic fields. In the current version, one can compute Hecke operators for principal ideals on these spaces, and determine the newforms. The algorithm, developed by Gunnells and Yasaki, is based on “Sharbly complexes”, and involves computations with the Voronoi polyhedron of the given imaginary field. – Construction of cuspidal spaces of modular forms over any imaginary quadratic field, of given level and weight 2 – Computation of dimensions – Computation of Hecke operators – Computation of new subspaces

16.10 Admissible Representations of GL2(Qp) This package, included since V2.16, provides functionality for working with the local components of an automorphic representation associated to a modular form. An eigenform in a classical space of cusp forms determines an automorphic representation, which is made up of “local components” at each prime p. The local component at p is an admissible representation on GL2(Qp). The package has two main parts. Starting with a cuspidal newform, the admissible representation (or data describing it) can be computed. Furthermore, via the local Langlands correspondence, there exists a related Galois representation on the absolute Galois group of Qp. The interesting part of this representation can be computed from the admissible representation. 98 Magma Overview V2.18 (December 16, 2011)

– Construction of the admissible representation on GL2(Qp) arising from a given cuspidal eigenform, of any weight and level – Conductor and central character of an admissible representation – Determination of a twist with minimal level, of a given representation – Recognition of whether a representation is principal series, or supercuspidal – In the case of principal series, the “principal series parameters” can be determined (as two characters on Zp) – In the supercuspidal case, a “cuspidal inducing datum” can be obtained (as a representation on a

suitable subgroup of GL2(Qp)) – Computation of the associated local Galois representation: its restriction to inertia is returned as a

representation on a ﬁnite group Gal(L/Qp) where L is some ﬁnite extension of Qp. Magma Overview V2.18 (December 16, 2011) 99

17 Diﬀerential Galois Theory

The Galois theory of linear differential equations is the analogue of the classical Galois theory of polynomial equations for linear differential equations. The natural analogue of the field in the classical case is the differential field. This is a field equipped with a derivation. We have constructed a basic facility for differential fields and rings. These types can be built from the algebraic function field or affine algebra types. Our medium term goal is to construct a fast solver for linear differential equations.

17.1 Diﬀerential Rings and Fields

– Construction of the rational differential field and the more general differential ring – Coercions, arithmetic and functionality for elements as for the underlying ring. – Changing the derivation of a differential ring. – Extending the constant ring of a differential ring – Wronskian matrix and Wronskian determinant – The differential constant field of a rational differential field – Ring and field extensions of differential rings and fields – Construction of a differential ideal – Quotient rings, rings and field of fractions of differential rings and fields

17.2 Diﬀerential Operator Rings

– Creation of a differential operator ring – Coercion, arithmetic and simple predicates for elements – Accessing coefficients of elements – Changing the derivation of a differential operator ring. – Changing the operator ring by extending the constant ring – Making a differential operator monic. – Adjoint of an operator – Applying an operator to an element of its basering – Euclidean algorithms, left and right (extended) GCD, (extended) left LCM – Companion matrix of an operator. – Determination of whether a place is regular, regular singular or irregular singular at an operator – All singular points of an operator – The indicial polynomial of an operator at a place – All rational solutions of an operator within a rational differential field – Newton polygon and Newton polynomial – Differential field extension of the base ring of an operator by adjoining a formal solution and formal derivatives – The symmetric power of an operator 100 Magma Overview V2.18 (December 16, 2011)

18 Geometry

The categories in this variety correspond to some classes of classical geometries.

• Finite planes • Incidence geometries • Polyhedra

18.1 Finite Planes Although ﬁnite planes correspond to particular families of designs, separate categories are provided for both projective and aﬃne planes in order to exploit the rich structure possessed by these objects.

– Creation of classical and non-classical finite projective and affine planes – Subplanes, dual of a projective plane – Numerical invariants: order, p-rank – Properties: Desarguesian, self-dual – Parallel classes of an affine plane – k-arcs: testing, complete, tangents, secants, passants – Conics: through given points, knot, exterior, interior – Unitals: testing, tangents, feet – Affine to projective planes and vice versa – Related structures: design, incidence matrix, incidence graph, linear code – Collineation group, isomorphism testing (optimized algorithm for projective planes) – Central collineations: testing, groups – Group actions on a plane: orbits and stabilizers of points and lines – Symmetry properties: point transitive, line transitive

Apart from elementary invariants, a reasonably fast method is available for testing whether a plane is desarguesian. Among special configurations of interest, a search procedure for k-arcs is provided. A specialized algorithm developed by Jeff Leon is used to compute the collineation group of a projective plane while the affine case is handled by the incidence structure method. The collineation group (order 2338) of a “random” projective plane of order 81 supplied by Gordon Royle was found in 1 202 seconds. As with graphs and designs the G-set mechanism gives the action of the collineation group on any appropriate set. Magma Overview V2.18 (December 16, 2011) 101

18.2 Incidence Geometry Magma contains facilities for creating and computing with incidence geometries and coset geometries. These have been developed by Dimitri Leemans (Brussels). The Magma Incidence Structure type comprises a set of points and a set of blocks together with an incidence relation. Following Bekenhout, we define a more general object as follows: An incidence geometry is a 4-tuple Γ = (X, ∗, t, I) where • X is a set of elements; • I is a non-empty set whose elements are called types; • t : X → I : x 7→ t(x) is a type function which maps an element to its type; • ∗ is an incidence relation that is a reflexive and symmetric relation such that ∀x, y ∈ X, x ∗ y and t(x) ∗ t(y) ⇒ x = y. We also introduce group-geometry pairs or coset geometries. Roughly speaking, these are geometries constructed from a group and some of its subgroups in the following way. Let I be a finite set and let G be a group together with a finite family of subgroups (Gi)i∈I . We define the incidence geometry Γ = Γ(G, (Gi)i∈I ) as follows. The set X of elements or varieties of Γ consists of all cosets gGi, g ∈ G, i ∈ I. We define an incidence relation * on X by:

g1Gi * g2Gj iﬀ g1Gi ∩ g2Gj is non-empty in G.

Γ(G, (Gi)i∈I ) may also be called a group-geometry pair.

18.2.1 Incidence Geometries

– Creation of incidence geometries – Conversion of an incidence geometry to a coset geometry – Set of types, rank – Diagram, incidence graph, elements – Residue, truncation, shadow, shadowspace – Properties: ﬂag-transitive geometry, residually connected, ﬁrm, thin, thick – Test if a geometry is a graph and conversion of such a geometry to a graph – Automorphism group – Correlation group

18.2.2 Coset Geometries

– Creation of coset geometries – Conversion of an incidence geometry to a coset geometry – Set of types, rank – Diagram – Residue, truncation – Properties: ﬂag-transitive geometry, residually connected, ﬁrm, thin, thick – Test if a geometry is a graph and conversion of such a geometry to a graph – Borel subgroup, group of the geometry – Maximal and minimal parabolic subgroups – Kernel of a geometry, i-kernels, quotient – Determine intersection properties – Test primitivity properties: primitive, weakly primitive, residually primitive, weakly residually primitive – Determine whether a coset geometry is locally 2-transitive 102 Magma Overview V2.18 (December 16, 2011)

18.3 Cones and Polyhedra A cone (in a toric lattice L) is the convex hull of finitely many rays. A (rational) polytope is the convex hull of finitely many points of L. More generally a (rational) polyhedron is given by the Minkowski sum of a polytope and a cone. There exist dual definitions in terms of the intersection of finitely many half-spaces. There is no requirement that polyhedra are of maximum dimension in the ambient toric lattice. Important operations include: – Standard constructions of cones, half-spaces, cross polytopes, cyclic polytopes, etc.; – Creation of cones and polyhedrons via a system of inequalities; – The defining inequalities of a cone or polyhedron; – The dual cone or polyhedron; – Taking the (affine) linear space spanned by, or contained in, a polyhedron; – Standard properties such as dimension, volume, and boundary volume; – Triangulations of a polytope and of its boundary; – Information on whether a polyhedron is simplicial, reflexive, terminal, etc.; – Construction of the cone in L × Z spanned by a polyhedron P in L; – The polyhedron defined by taking the convex hull of P ∩ Zn for any polyhedron P ; – The infinite and compact parts of a polyhedron; – Calculation of intersections, Minkowski sums, translations, etc.; – Taking hyperplane slices; – The minimum Q-generators and Z-generators of a cone; – Point membership; – Point counting and enumeration of lattice points; – The Ehrhart series, Ehrhart polynomial, and δ-vector of a polytope; – Faces of a cone or polyhedron, the f-vector and h-vector; – Construction of the face-poset; – The group of lattice automorphisms fixing a polytope. Magma Overview V2.18 (December 16, 2011) 103

19 Combinatorial Theory

This section is concerned with those topics in classical Combinatorial Theory that are supported by Magma. These include:- • Enumerative combinatorics • Partitions and tableaux • Graphs—directed and undirected • Networks • Incidence structures • Designs Other topices closely related to Combinatorial Theory, that are included in Magma, are Finite Fields, Finite Geometry, and Coding Theory. Descriptions for the facilities for these areas will be found in other sections.

19.1 Enumerative Combinatorics

– Factorial – Binomial, multinomial coeﬃcients – Stirling numbers of the ﬁrst and second kind – Fibonacci numbers, Bernoulli numbers, Harmonic numbers, and Eulerian numbers – Number of partitions of n – Enumeration of restricted and unrestricted partitions – Sets of subsets, multisets, and subsequences of sets – Permutations of sets (as sequences)

19.2 Partitions and Young Tableaux Extensive facilities were installed for working with Young tableaux and symmetric functions. The facility is heavily based on the Symmetrica package developed by A. Kerber and associates in Bayreuth. A highlight of the tableau machinery is the Robinson-Schensted-Knuth (RSK) correspondence. The collection of all symmetric functions deﬁned over some ring is viewed as forming an algebra where the chief interest lies in the change of basis matrix relative to diﬀerent bases. The Young tableaux code was written by G. White while the machinery for symmetric functions was adapted from Symmetrica by A. Kohnert (Bayreuth) during a year-long visit to Sydney.

– Number of partitions of n – Enumeration of restricted and unrestricted partitions – Conjugate partitions – Longest increasing sequences within words of integers – Lexicographical ordering of words – Creation of skew or non-skew tableaux over the integers or arbitrary label sets – Enumeration of tableaux – Random tableaux – Properties: shape, skew-shape, weight, hooklength’s, content, row/column words – Operations: diagonal sum, product, conjugate, jeu de taquin, row insertion, inverse jeu de taquin, inverse row insertion – RSK correspondence, inverse RSK correspondence – Enumeration of tableaux having speciﬁed size and alphabet 104 Magma Overview V2.18 (December 16, 2011)

19.3 Symmetric Function Algebras Symmetric functions are defined as polynomials in an infinite number of variables, invariant under the action of the symmetric group. The set of all symmetric functions denoted by Λ is an algebra. A symmetric function algebra may be created over an arbitrary commutative ring (with unity) R. Five different bases for an algebra of symmetric functions are supported. – Creation of algebras with any one of five possible bases: consisting of Schur functions, Elementary, Monomial, Homogeneous (complete) or Power Sum functions. – Creation of elements as linear combinations of basis elements (indexed by partitions) or from coercing a polynomial or a scalar. – An algebra can be tested for which basis it represents its elements with respect to. – Alternative print styles for elements of an algebra – Addition, subtraction, multiplication and composition (plethysm) of symmetric functions. Testing for homogeneity and equality. – Decomposition of symmetric functions into basis elements and coefficients thereof. – Symmetric functions can be coerced into polynomial rings. – Frobenius homomorphism, inner product, tableax with a given generating function and corresponding characters – Matrices converting from any of the five bases to any other of the five bases Magma Overview V2.18 (December 16, 2011) 105

19.4 Graphs Graphs may be directed or undirected. In addition, their vertices and/or their edges may be labelled. A graph may be represented by means of its adjacency matrix, or by means of its adjacency list.

– Directed and undirected graphs – Optional vertex and edge labels – Operations: union, join, product, contraction, switching, etc. – Standard graphs: complete, complete bipartite, k-cube – Properties: connected, regular, eulerian – Algebraic invariants: Characteristic polynomial, spectrum – Diameter, girth, circumference – Connectedness, paths and circuits – Triconnectivity – General vertex and edge connectivity in graphs and digraphs – Maximum matching in bipartite graphs – Spanning trees, cut-vertices – Cliques and maximal cliques, independent sets – Chromatic number, chromatic index, chromatic polynomial – Planarity testing, obstruction isolation, faces, embedding – Graphs from groups: Cayley graph, orbital graph – Automorphism group (B. McKay’s algorithm), canonical labelling – Testing of pairs of graphs for isomorphism – Group actions on a graph: orbits and stabilizers of vertex and edge sequences – Symmetry properties: vertex transitive, edge transitive, k-arc transitive, distance transitive, distance regular – Intersection numbers of a distance regular graph – Interface for B. McKay’s graph generation algorithm [32]

Brendan McKay’s automorphism program nauty [33] is used for computing automorphism groups and for testing pairs of graphs for isomorphism. In accordance with the Magma philosophy, a graph may be studied under the action of an automorphism group. Using the G-set mechanism an automorphism group can be made to act on any desired set of objects derived from the graph. The planarity tester and obstruction isolator are linear time algorithms; they are due to Boyer and Myrvold [5]. The triconnectivity algorithm is the classical linear time algorithm from Hopcroft and Tarjan [22] with corrections of our own and from Gutwenger and Mutzel [20]. The general vertex and edge connectivity machinery rests on two ﬂow-based algorithms for networks: the Dinic algorithm and the push-relabel method (see the section on Networks below). 106 Magma Overview V2.18 (December 16, 2011)

19.5 Multigraphs A multigraph is a graph which may contain loops or multiple (parallel) edges between the same pair of vertices. Vertices and edges in a multigraph may be labelled, and edges may also have associated capacities or weights, to assist ﬂow-based or shortest-path based algorithsm. Multigraphs are represented by means of an adjacency list. Much of the same machinery is available for simple graphs and for multigraphs.

– Directed and undirected multigraphs – Optional vertex and edge labels, capacities, weights – Conversion to and from simple graphs – Operations: union, join, product, contraction, etc. – Properties: bipartite, regular, connected, biconnected – Triconnectivity – General vertex and edge connectivity – Maximum matching in bipartite graphs – Spanning trees, cut-vertices, shortest paths – Planarity testing, obstruction isolation, faces, embedding See the section on Graphs (above) for comments on some of the algorithms.

19.6 Networks A network is a multidigraph whose edges are associated with a capacity. It may have multiple (parallel) edges and loops. A network is represented by means of its adjacency list.

– Operations: union, join, contraction – Maximum ﬂow and minimum cut

There are two implementations of the maximum ﬂow algorithm. One is the classical Dinic algorithm with heuristics due to McKay, the other is a push-relabel method with heuristics mainly due to Cherkassky and Golberg et al. [12, 13]. Magma Overview V2.18 (December 16, 2011) 107

19.7 Incidence Structures and Designs General incidence structures provide a universe in which families of incidence structures satisfying stronger conditions (linear spaces, t-designs, etc.) reside.

– Creation of a general incidence structure, near-linear space, linear space, design – Diﬀerence sets: standard diﬀerence sets, development – Hadamard designs, Witt designs – Unary operations: complement, contraction, dual, residual – Binary operations: sum, union – Invariants for an incidence structure: point degrees, block degrees, covalence – Invariants for a design: replication number, order, covalence, intersection numbers, Pascal triangle – Properties: balanced, complete, uniform, self-dual, simple, Steiner – Near-linear space operations: connection number, point and line regularity, restriction – Resolutions, parallelisms, parallel classes – Graphs and codes from designs: block graph, incidence graph, point graph, linear code – Automorphism group (J. Leon’s algorithm), isomorphism testing – Group actions on a design: orbits and stabilizers of points and blocks – Symmetry properties: point transitive, block transitive

Tools are provided for constructing designs from diﬀerence sets, Hadamard matrices, codes and other designs. The standard families of diﬀerence sets are incorporated. A major feature is the ability to compute automorphism groups and to test pairs of incidence structures for isomorphism.

19.8 Hadamard Matrices Hadamard matrices are square matrices whose entries are all ±1, such that any two rows (or columns) are orthogonal. They have some useful special properties; in particular, they lead to certain classes of design; produce a family of error-correcting codes that can correct many errors; and have several important applications in signal processing. – Checking of the Hadamard property – Normalised form, simple invariant testing (4-profile) – Canonical form, equivalence testing – Construction of 3-designs from rows or columns – Automorphism group – Database of Hadamard and skew Hadamard matrices; an interface for augmenting this database with new matrices is provided The problem of determining if two Hadamard matrices are equivalent is a difficult one. The computation of a canonical form (due to Brendan McKay’s program nauty [33]) is one way to achieve this; Magma also contains an older algorithm of Jeff Leon that performs this equivalence testing. Optional databases of Hadamard and skew Hadamard matrices are available. The former contains 5 391 examples of inequivalent Hadamard matrices for all possible degrees up to 256 (the degree must be 1, 2, or a multiple of four), and is a complete listing for degree up to 28. The skew Hadamard database contains 638 inequivalent skew Hadamard matrices of degree 36, 44, or 52. 108 Magma Overview V2.18 (December 16, 2011)

20 Coding Theory

Magma provides support for several diﬀerent branches of Coding Theory. These include:-

• Linear codes over finite fields • Algebraic-geometric codes • Low density parity check codes • Linear codes over Galois rings • Additive codes over finite fields • Quantum stabilizer codes

20.1 Linear Codes over Finite Fields 20.1.1 Linear Codes: Creation

– Creation from a subspace of a vector space – Creation in terms of a generator matrix – Creation from a design – Creation from a ﬁnite plane – Construction of a cyclic code given the generator polynomial – Construction of a cyclic code given the roots of the generator polynomial – Universe code, repetition code, zero code – Random linear code

20.1.2 Operations on Codewords

– Vector space operations: sum, diﬀerence scalar multiplication – Syndrome – Distance and weight – Coordinates, support – Trace

20.1.3 Elementary Operations

– Sum, intersection – Dual code – Hull of a code – External direct sum, Plotkin sum – Modifying the codewords: augment, extend, expurgate, lengthen, puncture, shorten, etc. – Subcode generated by given codewords – Subcode of a speciﬁed dimension – Subcode generated by words of a given weight – Coset leaders (in the case of a small code) Magma Overview V2.18 (December 16, 2011) 109

20.1.4 Basic Properties – Standard form – All information sets of a code – Idempotent of a cyclic code – Properties: cyclic – Properties: even, doubly even, equidistant – Properties: self-dual, weakly self-orthogonal – Properties: perfect, nearly perfect – Properties: maximum-distance separable, equidistant – Properties: optimal for the Griesmer bound – Properties: projective – Determine whether two codes are equivalent

20.1.5 Weight Distribution – Minimum weight (Zimmermann algorithm) – Weight distribution, weight enumerator – MacWilliams transform – Complete weight enumerator, MacWilliams transform – Number of words of designated weight – Number of words of constant weight – List all words of designated weight – List all words of constant weight – Coset weight distribution, covering radius, diameter

Carefully crafted algorithms are provided for computing the minimum weight, the weight distribution, and word collection algorithms. For example, computation of the minimum weight of a [96, 60, 8] code takes 77 seconds; computation of the weight distribution of the [64, 22, 16] Reed-Muller code (r = 2, m = 6) takes 1.4 seconds.

20.1.6 Construction of Families – Hamming code, simplex code – Reed-Muller codes – Quadratic residue code, Golay codes, doubly circulant QR-code, twisted QR-code, power residue code – BCH code – Goppa code – Chien-Choy code – Alternant code, Fire code, Gabidulin code – Srivastava, generalized Srivastava codes – Reed-Solomon code, generalized Reed-Solomon code, Justesen code – Maximum distance separable code

20.1.7 Algebraic-Geometric codes Functions are provided for the construction of algebraic–geometric codes. The user chooses a plane curve X, and speciﬁes a set of places of degree 1 on X and a divisor on X. 110 Magma Overview V2.18 (December 16, 2011)

20.1.8 Decoding Algebraic-Geometric codes

Algebraic-geometric codes can be decoded eﬃciently up to the Goppa designated distance.

20.1.9 Changing the Alphabet

– Extension of the base field – Restricting the alphabet to a subfield – Subfield subcode – Trace code – Rewriting the alphabet, taken as the elements of GF (qm), as m-dimensional vectors over GF (q)

20.1.10 Combining Codes

– Concatenation of two codes – Concatenated code – Construction X – Construction X3 – Construction XX – Zinoviev code – Construction Y1

20.1.11 Bounds

– BCH bound on minimum distance – Upper and lower bounds on the cardinality of a largest code having given length and minimum distance – Upper asymptotic bounds on the information rate – Tables of best known bounds, based on the codes database

20.1.12 Best Known Codes

Magma incorporates databases containing constructions of the best known linear codes over Fq for q = 2, 3, 4, 5, 7, 8, 9. These databases contain codes of length up to 256 for q = 2, 4; of length up to 243 for q = 3; of length up to 130 for q = 5, 8, 9; and of length up to 100 for q = 7. The binary codes database is complete in the sense that it contains a construction for every set of parameters, with the codes of length up to 31 known to be optimal. The database for codes over F3 is more than 78% complete (and complete up to length 100), with codes of length up to 21 known to be optimal. The database for codes over F4 is more than 65% complete (and complete up to length 97), with codes of length up to 18 known to be optimal. Further details are summarised in the table below. The Magma BKLC database makes use of the tables of bounds compiled by A.E. Brouwer. It should be noted that the Magma BKLC database is unrelated to the similar (but rather incomplete) BKLC database forming part of GUAVA, a share package in GAP3. A signiﬁcant number of entries in the Magma BKLC database provide better codes than the corresponding ones listed in the Brouwer tables. The construction of the Magma BKLC database has been undertaken by John Cannon (Syd- ney), Markus Grassl (Karlsruhe) and Greg White (Sydney). Magma Overview V2.18 (December 16, 2011) 111

F2 F3 F4 F5 F7 F8 F9 nmax 256 243 256 130 100 130 130 nopt 31 21 18 15 14 14 16 ncomplete 256 100 97 80 68 76 93 total 33 152 29 889 33 152 8 645 5 150 8 645 8 645 missing 0 6 545 11 379 527 381 1 763 1 333 ﬁlled 100% 78.10% 65.67% 93.90% 92.60% 79.61% 84.58%

20.1.13 Decoding Algorithms – Syndrome decoding – Alternant decoding

20.1.14 Automorphism Group – Automorphism groups of linear codes over GF(p) (prime p), GF(4) – Testing pairs of codes for isomorphism over GF(p) (prime p), GF(4) – Group actions on a code

Automorphism groups may be computed over any ﬁeld Fp, p a prime, and F4, again using Leon’s PERM package.

20.1.15 Attacks on the McEliece Cryptosystem All of the best known decoding attacks on the McEliece cryptosystem are available, as well as improved attacks. – McEliece’s attack – Lee and Brickell’s attack – Leon’s attack – Stern’s atack – Canteaut and Chabaud’s attack – Generalized combinations of attacks 112 Magma Overview V2.18 (December 16, 2011)

20.2 Low-Density Parity-Check Codes

Low-density parity-check (LDPC) codes are among the best performing linear codes in practice, being capable of correcting n errors, where n is close to the Shannon bound. Currently, there exist few techniques for the explicit construction of LDPC codes. Typically, these codes are selected at random from an ensemble, and their properties determined through simulation. An LDPC code can be decoded in time that is linear in its block length through use of iterative belief propagation techniques.

– Construction of LDPC codes from sparse matrices – Gallager’s construction – The group-based construction of Margulis – Deterministic LDPC constructions – Random constructions from regular and irregular LDPC ensembles – Construction of the Tanner graph of an LDPC code – Ensemble rate – Iterative decoding of LDPC codes – Simulation of decoding performance on speciﬁed channels – Density evolution and threshold determination for binary symmetric and Gaussian channels for given channel parameters – A small database of good irregular LDPC ensembles is provided Magma Overview V2.18 (December 16, 2011) 113

20.3 Linear Codes over Finite Rings

In the 1990’s it was discovered that certain good nonlinear codes could be obtained by applying the Gray mapping to certain linear codes defined over the ring Z4. This led to a great deal of research into the properties of codes over finite rings. This section describes the facilities which are provided in Magma for studying linear codes over finite rings. Magma currently supports basic facilities for codes over integer residue rings and Galois rings, including the construction of cyclic codes, and the calculation of the complete weight enumerator. Additional functionality is available for the special case of codes over Z4.

20.3.1 Constructions – Creation from a subspace of a vector space – Creation in terms of a generator matrix – Construction from a permutation – Construction of a cyclic code given the generator polynomial – Construction of a cyclic code given the roots of the generator polynomial – Universe code, repetition code, zero code – Random linear code

– Construction of the Kerdock and Preparata codes (over Z4)

20.3.2 Elementary Operations and Properties – Syndrome – Hamming distance, Hamming weight

– Lee distance, Lee weight (over Z4) – Coordinates, support – Trace – Module operations: sum, diﬀerence, scalar multiplication – Dual code – External direct sum, Plotkin sum – Obtaining a new code by modifying codewords: Augment, extend, expurgate, lengthen, puncture, shorten, etc. – Coset leaders (in the case of a small code)

– Standard form, Gray map, Lee weights (over Z4)

– Determine whether cyclic, even, self-dual, self-orthogonal (over Z4)

20.3.3 Weight Distribution – Minimum Hamming weight – Hamming weight distribution – Hamming weight enumerator – Complete weight enumerator

– Minimum Lee weight via the Zimmermann algorithm (over Z4)

– Lee weight distribution (over Z4)

– Symmetric weight enumerator (over Z4)

– Lee weight enumerator (over Z4) 114 Magma Overview V2.18 (December 16, 2011)

20.4 Additive Codes

Given a finite field F and the space of all n-tuples of F , an additive code is a subset of F (n) which is a K-linear subspace for some subfield K ⊆ F . Additive codes have become increasingly important recently due to their application to the construction of quantum error-correcting codes, though they are also of interest in their own right. The Magma package for quantum error-correcting codes is built on the machinery for additive codes.

20.4.1 Constructions – Additive codes defined as some K-additive subspace of F n for some subfield K of F . – Creation in terms of a generator matrix – Construction of a cyclic code given the generator polynomial – Universe code, repetition code, zero code – Random additive code – Sum, intersection of two codes – Dual code, symplectic dual code – Plotkin sum – Obtaining a new code by modifying codewords: augment, extend, lengthen, puncture, shorten, etc. – Subcode generated by given codewords – Subcode of a specified dimension – Subcode generated by words of a given weight – Construction of a cyclic (quasicyclic) code from parameters

20.4.2 Operations and Properties – Vector space operations: sum, diﬀerence scalar multiplication – Distance and weight – Coordinates, support – Trace – Determine whether cyclic, self-dual, self-orthogonal, symplecic self-dual, symplectic self-orthogonal – Automorphism and permutation groups of additive codes over GF(4) – Group actions on a code

20.4.3 Weight Distribution – Minimum weight (Zimmermann algorithm) – Weight distribution, weight enumerator – MacWilliams transform – Complete weight enumerator, MacWilliams transform – Number of words of designated weight – Number of words of constant weight – List all words of designated weight – List all words of constant weight

The fast algorithm used for determining the minimum weight and word collection is adapted from the one used for linear codes and provides performance similar to a calculation with an equivalently-sized linear code. Magma Overview V2.18 (December 16, 2011) 115

20.5 Quantum Error-Correcting Codes

In 1995 Shor showed that it was possible to encode quantum information in such a way that errors can be corrected. Following this discovery, a class of quantum error-correcting codes known as stabilizer codes was developed. It turned out that these stabilizer codes can be represented by additive codes over ﬁnite ﬁelds. This remarkable result reduces the problem of constructing fault- tolerant encodings on a continuous Hilbert space to that of constructing certain discrete codes, thereby permitting the application of many of the tools developed in classical coding theory.

20.5.1 Constructions – Quantum stabilizer codes deﬁned by symplectic self-dual additive codes. – Creation in terms of a generator matrix – Construction of a cyclic code given the generator polynomial – Universe code, repetition code, zero code – Random quantum code – Construction of new codes via direct sums – Construction of new codes extending or shortending existing codes – Families: Cyclic codes, quasicyclic codes, CSS code,

20.5.2 Quantum Codes: Basic Properties – Properties: Cyclic, quasicyclic – Properties: Self-dual – Properties: Stablizer code – The group of errors on a quantum space – Stabiliser group associated with a quantum code – Minimum weight (Zimmermann algorithm) – Weight distribution, weight enumerator – Complete weight enumerator – Automorphism and permutation groups of binary quantum codes (codes based on quaternary stabilizer codes)

The algorithm for determining the minimum weight applies the Zimmermann algorithm to the underlying self-dual code. 116 Magma Overview V2.18 (December 16, 2011)

21 Cryptography 21.1 Pseudo-Random Sequences Magma provides tools for the creation and analysis of pseudo-random bit sequences. The universe of these sequences is generally GF(2). However, some functions, such as Berlekamp-Massey, apply to sequences defined over arbitrary finite fields.

– Generation of n elements of a Linear Feedback Shift Register sequence (LFSR sequence) – Next state of a LFSR sequence – Characteristic polynomial of a LFSR sequence (BerlekampMassey algorithm) – Shrinking generator – Random sequence via the RSA random bit generator – Random sequence via the Blum Blum Shub generator – Auto-correlation of a binary sequence – Cross correlation of two binary sequences – Decimation of a sequence Magma Overview V2.18 (December 16, 2011) 117

22 Mathematical Databases

Magma includes a growing number of mathematical databases. Typically, such a database contains a complete classiﬁcation of all structures of some given type up to a speciﬁed bound. A number of these databases are an integral part of algorithms installed in Magma. The current databases include:

22.1 Algebraic Geometry

• Cremona Database of Elliptic Curves: A database constructed by John Cremona that contains all isogeny classes of elliptic curves having conductor up to 130,000 is available.

• Stein-Watkins Database of Elliptic Curves: The Stein-Watkins database of 136,924,520 elliptic curves of conductor up to 108 is now available in Magma.

• K3 Surfaces: This comprises a collection of 24,099 K3 surfaces. For g = −1, 0, 1, 2, all K3 surfaces of genus g are included, there being 4281, 6479, 6627 and 6628 surfaces, respectively. For higher genus, the data associated to the 6628 K3 surfaces of genus 2 propagates in a predictable way, so only those K3 surfaces with codimension at most 7 and genus in the range 3 to 9 are included.

• 3-folds: Basic machinery is provided that allows the user to generate lists of Fano 3-folds and Calabi–Yau 3-folds.

22.2 Coding Theory

• Best Known Binary Linear Codes: A database containing constructions of the best known linear codes over F2 of length up to 256 has been implemented by M. Grassl and the Magma group from tables of A. E. Brouwer. The codes of length up to 31 are optimal. The database is complete in the sense that it contains a construction for every set of parameters. Thus the user has access to 33 152 binary codes.

• Best Known F3 Linear Codes: A database containing constructions of the best known linear codes over F3 of length up to 243. This database has been constructed by M. Grassl and the Magma group. The codes of length up to 21 are optimal.

• Best Known F4 Linear Codes: A database containing constructions of the best known linear codes over F4 of length up to 256. This database has been constructed by M. Grassl and the Magma group. The codes of length up to 18 are optimal.

• Best Known F5,F7,F8, and F9 Linear Codes: Similar databases exist for these codes. Their details and more information about the previous ones are summarised in the table below.

22.3 Finite Fields • Irreducible Polynomials: A database of sparse irreducible polynomials, constructed by Allan Steel in 2004–2007. The polynomials have the form f(x) = xn +g(x), where the degree of g is minimal and g is the first such polynomial in lexicographical order. The database has the following degrees for the specified fields:

– GF(2): up to degree 120,000. – GF(3): up to degree 50,000. – GF(4), GF(5), GF(7): up to degree 20,000. – GF(q), 9 ≤ q ≤ 127: up to degree 1,000 or more.

• Conway Polynomials: A database of Conway polynomials for F2 to F127. These polynomials are primitive and provide standard definitions for finite field extensions (used in modular representation theory, for example). This database is based on lists built by Richard Parker and Frank Lübeck. The database now extends up to degree 409 for p = 2 (with some gaps) and up to degree at least 4 for all p ≤ 109987.

22.4 Graph Theory • Small Graphs: Several databases of small graphs are available, using data made available by Brendan McKay. These include:-

– All simple graphs on 2 to 10 vertices – All connected simple graphs on 2 to 10 vertices – All Eulerian graphs on 2 to 12 vertices – All connected Eulerian graphs on 3 to 11 vertices – Planar graphs on up to 11 vertices, – Some self-complementary graphs of orders up to 20.

• Simple Graphs: Magma contains an interface to the graph enumeration program of Bren- dan McKay which allows the user to rapidly construct all simple graphs on a given number of vertices. The graph generation program allows the user to specify one or more conditions thereby allowing the construction of all graphs on a given number of vertices satisfying the condition.

• Strongly Regular Graphs: A database containing a list of strongly regular graphs constructed by Brendan McKay, Ted Spence and others. This database contains strongly regular graphs on 25, 26, 27, 28, 29, 35, 36, 37 and 40 vertices. The graphs are indexed by the order of the graph, its degree, the number of common neighbours to each pair of adjacent vertices, and the number of common neighbours to each pair of non-adjacent vertices.

22.5 Group Theory • Small Groups: The Small Groups Library developed by Besche, Eick and O’Brien. This database contains all groups of order up to 2000, except the groups of order 1024, and a number of inﬁnite series of larger groups.

• The ATLAS Database: Representations of nearly simple groups, as in the Birmingham ATLAS of Finite Group Representations. The data was supplied by Rob Wilson. Magma Overview V2.18 (December 16, 2011) 119

• Almost Simple Groups: This database contains information about almost simple groups G, where S ≤ G ≤ Aut(S) and S is a simple group. The groups G that are included in the database are those associated with S such that |S| is less than 16000000, as well as M24, HS, J3, McL, Sz(32) and L6(2). In particular, the database includes all simple groups having a permutation representation of degree less than 1000. The groups in the database are deﬁned on standard generators which can be used to create an isomorphism between an almost simple group in some arbitrary representation and the “standard” version of it stored in the database. The database was originally conceived by Derek Holt with a major extension by Volker Gebhardt and sporadic additions by Bill Unger. The following groups are included:

– Alternating groups: An for n ≤ 999;

– Classical groups: L2(q), L3(q), L4(q) and L5(q) for all prime powers q; L6(3), L7(3); Ld(2) for d ≤ 14; Sp4(q) for all odd prime powers q and even q ≤ 16; Sp6(3), Sp8(2), Sp10(2); U3(q) for all primes q and prime powers q ≤ 25; U4(q) for prime powers q ≤ 7, U6(2); ± m ± O7(3), O8 (2), O8 (3), O10(2)

– Exceptional groups: G2(3), G2(4), G2(5) 3 2 0 – Twisted groups: Sz(8), Sz(32), D4(2), F4(2)

– Sporadic groups: M11, M12, M22, M23, M24, J1, J2, J3, HS, McL, He, Co2, Co3, F i22

• Simple Groups: A database containing a presentation, the conjugacy classes and maximal subgroups for each simple group of order less than a million. The database was prepared by Jamali, Robertson and Campbell.

• Perfect Groups: The database of perfect groups of order up to a million constructed by Holt and Plesken.

• Transitive Groups: The transitive permutation groups of degree up to 30. The transitive groups of degree up to 15 were determined by Butler and McKay while the classiﬁcation was extended to degree 30 by Hulpke.

• Primitive Groups: The table of primitive groups of degree up to 2499 (Sims, Roney-Dougal & Unger, Roney-Dougal).

• Permutation Representations: A collection of ﬁnite groups given in terms of permutation representations. A particular group is included if:

– It is an ‘interesting’ group. In practice this means a sporadic simple group or a close relative of such; or – It is representative of some class of groups which is useful for testing conjectures and algorithms.

• Irreducible Matrix Groups: The table of irreducible subgroups of GL(n, p) where p is prime and pn ≤ 2499 (Sims, Roney-Dougal & Unger, Roney-Dougal).

• Irreducible Soluble Groups: The irreducible soluble subgroups of GL(n, p) for n > 1 and pn < 256, as classiﬁed by Short.

• Finite Groups of Integral Matrices: This database contains representatives for all GLn(Z)-conjugacy classes of irreducible maximal ﬁnite subgroups of GLn(Z) for n <= 11 and n ∈ {13, 17, 19, 23}.

• Finite Groups of Rational Matrices: The maximal ﬁnite subgroups of GL(n, Q), for n up to 31. 120 Magma Overview V2.18 (December 16, 2011)

• Symplectic Matrix Groups: The maximal ﬁnite irreducible subgroups of Sp2n(Q) for 1 ≤ i ≤ 11. These groups were classiﬁed by Markus Kirschmer who also constructed the database.

• Quaternionic Matrix Groups: The ﬁnite absolutely irreducible subgroups of GLn(D) where D is a deﬁnite quaternion algebra whose centre has degree d over Q and nd ≤ 10

22.6 Hadamard Matrices • Hadamard Matrices: This database includes all inequivalent Hadamard matrices of degree at most 28, and examples of matrices of all degrees up to 256. The matrices up to degree 28 have been provided by Neil Sloane while most of those of larger order have been provided by C. Koukouvinos, I. Kotsireas and G. Stelios. Several matrices of dimensions between 36 and 60 were provided by D. Djokovic, including 1771 of degree 60.

• Skew Hadamard Matrices: This database includes known skew Hadamard matrices of degree up to 256. The matrices up to degree 28 have been provided by Neil Sloane while most of those of larger order have been provided by C. Koukouvinos, I. Kotsireas and G. Stelios.

22.7 Lattices • Database of Interesting Lattices: A database containing lattices of Sloane and Nebe, containing the automorphism group and Θ-series for many examples.

22.8 Lie Algebras • Solvable Lie algebras: All soluble Lie algebras of dimensions 2, 3 and 4 over any ﬁeld are included in this database which was constructed by Willem de Graaf.

• Nilpotent Lie algebras: This comprises all nilpotent Lie algebras having dimensions 3, 4, 5, and 6 over all base ﬁelds, except base ﬁelds of characteristic 2, when the dimension is 6. This database is due to Willem de Graaf.

22.9 Number Theory • Cunningham Factorizations: A database containing 237,578 factors f of numbers an ±1, where a < 10000, n < 10000, and f > 109. The factorizations of integers of the form an ± 1, a ≤ 12, were produced by Sam Wagstaﬀ and collaborators (for n up to 1200), with contributions from Arjen Bot, Will Edgington, Alexander Kruppa and Paul Leyland (for larger n); for 13 < a < 99 they are mainly from the Brent-Montgomery-te Riele extension of the Cunningham tables, with contributions by ECMNET and various individuals; for 100 < a < 1000 they are mainly from the tables produced by Hisanori Mishima and Mitsuo Morimoto with additions from Rob Hooft, Pete Moore and others. For prime bases a < 1000 Richard Brent computed many of the factors for an unpublished extension of the Brent- Montgomery-te Riele tables.

• Galois Polynomials: For each transitive group G with degree between 2 and 15, the database contains a univariate polynomial over the integers which has G as its Galois group. These polynomials have been determined by J. Kl¨unersand G. Malle.

• Number Fields: Databases of some 2.6 million small number ﬁelds of degrees two to nine. These databases are searchable on discriminant range, signature, class number, class group, and Galois group. Magma Overview V2.18 (December 16, 2011) 121

• Function Fields: Databases of various function fields with specified base finite field and degree. Supported combinations are: F2, degrees 2 and 3; F3, degree 2, F4, degrees 2, 3, and 5; F5, degrees 2, 3, 4, and 8; F7, degree 9; F11, degree 3; and F13, degree 3.

22.10 Topology • Fundamental Groups of 3-manifolds: A database containing the 11,126 small-volume closed hyperbolic manifolds of the Hodgson-Weeks census. Each manifold record contains a presentation of the fundamental group and a homomorphism to Sn whose kernel has positive betti number. 122 Magma Overview V2.18 (December 16, 2011)

23 Documentation

Magma has an extensive online help system. It includes specially-written material on all aspects of Magma, suitable for the ﬁrst-time user, and complete access to the Magma Handbook which provides a full description of all the facilities of the system. The help nodes are structured as a tree, allowing a full-scale browsing facility in addition to simple help requests. In addition to the online help, there are ﬁve main components to the printed documentation: – J. Cannon and C. Playoust: First Steps in Magma, 16 pages – J. Cannon and C. Playoust: An Introduction to Algebraic Programming with Magma: The Language, 350 pages – W. Bosma, J. Cannon and C. Playoust: An Introduction to Algebraic Programming with Magma: The Categories, 500 pages – W. Bosma and J. Cannon: The Magma Handbook, Eight Volumes, 3100 pages. – W. Bosma, J. Cannon et al.: Solving Problems with Magma, 220 pages

References [1] Selma Altınok, Gavin Brown, and Miles Reid. Fano 3-folds, K3 surfaces and graded rings. In Topology and geometry: commemorating SISTAG, volume 314 of Contemp. Math., pages 25–53. Amer. Math. Soc., Providence, RI, 2002. [2] R. Beals and C. R. Leedham-Green and A. C. Niemeyer and C. E. Prager and A. Seress, A black-box algorithm for recognising finite symmetric and alternating groups. Trans. Amer. Math. Soc. to appear. [3] K. Belabas, M. van Hoeij, J Klüners,A. Steel, Factoring polynomials over global fields, Journal de Théoriedes Nombres de Bordeaux, 21 (2009), 15–39. [4] B. J. Birch. Hecke actions on classes of ternary quadratic forms. Computational number theory (Debrecen, 1989), 191–212, de Gruyter, Berlin, 1991. [5] J. Boyer and W. Myrvold. Simplified O(n) Planarity Algorithms. Submitted, 2001. [6] W. Bosma, J. J. Cannon and A. Steel. Lattices of Compatibly Embedded Finite Fields, J. Symb. Comp. 24 (1997), pp. 351–369.

[7] S. Bratus and I. Pak. Fast constructive recognition of a black box group isomorphic to Sn or An using Goldbach’s conjecture. J. Symb. Comput. 29 (2000), pp. 33–57. [8] Gavin Brown. Datagraphs in algebraic geometry. In Symbolic and Numerical Scientific Com- puting – Proc. of SNSC’01, volume 2630 of LNCS, Heidelberg, 2003. Springer-Verlag. F. Winkler and U. Langer (eds.). [9] J. Buchmann and D. Squirrel. Kernels of Integer Matrices via Modular Arithmetic. Technical Report No. TI-4/99, University of Darmstadt. . [10] J. Buchmann, M.J. Jacobson Jr., and E. Teske. On Some Computational Problems in Finite Abelian Groups. Mathematics of Computation, 66:1663–1687, 1997. [11] Anita Buckley and BalázsSzendrHoi. The Riemann–Roch formula for 3-folds with canonical singularities and applications to Calabi–Yau 3-folds. 19pp preprint as math.AG/0309033, 2002 [12] B.V. Cherkassky and A.V. Golberg. On Implementing the Push-Relabel Method for the Max- imum Flow Problem. Algorithmica, 19:390–410, 1997. [13] B.V. Cherkassky, A.V. Golberg, Paul Martin, J.C. Setubal, and J. Stolfi. Augment or Push? A Computational Study of Bipartite Matching and Unit Capacity Flow Algorithms. Technical report, NEC Research Institute, 1998. Magma Overview V2.18 (December 16, 2011) 123

[14] J. Cremona. Algorithms for modular elliptic curves. Second edition. Cambridge University Press, Cambridge, 1997. [15] H. Derksen. Computation of Invariants for Reductive Groups, Adv. Math 141 (1999), 366– 384. [16] J. Della Dora, C. Dicrescenzo and D. Duval. About a new method for computing in algebraic number fields, Proc. EUROCAL ’85, Vol. 2. Lecture Notes in Computer Science (1985), 204, 289–290. [17] Dominique Duval. Rational Puiseux Expansions. Compositio Mathematica, 70:119–154, 1989. [18] M. Eichler. The basis problem for modular forms and the traces of the Hecke operators. Modular functions of one variable, I, 75–151, Lecture Notes in Math., 320, Springer, Berlin, 1973. [19] V. Gebhardt. Constructing a short defining set of relations for a finite group. J. Algebra 233 (2000), pp. 526–542. [20] C. Gutwenger and P. Mutzel. A Linear Time Implementation of SPQR-Trees. In J. Marks, editor, Graph Drawing 2000, volume 1984 of Lecture Notes in Computer Science, pages 70–90. Springer-Verlag, 2001.

[21] H. Hijikata, A. Pizer, T. Shemanske. The basis problem for modular forms on Γ0(N). Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), no. 6, 280–284. [22] J.E. Hopcroft and R.E. Tarjan. Dividing a Graph into Triconnected Components. SIAM J. Comput., 2(3):135–158, 1973. [23] W. Kantor and A. Seress. Black box classical groups. Mem. Amer. Math. Soc. 149 (2001), no. 708. [24] G. Kemper and A. Steel. Some Algorithms in Invariant Theory of Finite Groups, Proceedings of the Euroconference on Computational Methods for Representations of Groups and Algebras, P. Dr¨axler,G.O. Michler, and C.M. Ringel, eds., Birkh¨auser,Basel, 1998. [25] G. Kemper. An Algorithm to Calculate Optimal Homogeneous Systems of Parameters, JSC 27 2 (1999), 171–184 [26] D. R. Kohel. Hecke module structure of quaternions. Class Field Theory – it’s Centenary and Prospect, K. Miyake, ed., The Advanced Studies in Pure Mathematics Series, Math. Soc. Japan, 2001.

[27] D. R. Kohel and W. A. Stein. Component groups of quotients of J0(N). Proceedings of ANTS IV, 2000. [28] B. A. LaMacchia and A. M. Odlyzko. Solving Large Sparse Linear Systems over Finite Fields, Advances in Cryptology: Proceedings of Crypto ’90, A. Menezes, S. Vanstone, eds., Lecture Notes in Computer Science 537, Springer-Verlag, NY (1991), 109–133. [29] C. R. Leedham-Green and S. H. Murray. Variants of product replacement. Contemp. Math. 298 (2002), pp. 97–104. [30] E. M. Luks and A. Seress. Computing the Fitting subgroup and solvable radical of small-base permutation groups in nearly linear time. Groups and computation II (New Brunswick, NJ, 1995), 169–181, Amer. Math. Soc., 1997. [31] J.-F. Mestre. Sur la m´ethode des graphes, Exemples et applications, Proceedings of the international conference on class numbers and fundamental units of algebraic number ﬁelds, Nagoya University, 1986, 217–242. [32] B. D. McKay. Isomorph-free exhaustive generation. J. Algorithms, 26 (1998), 306–324. [33] B. D. McKay. Practical Graph Isomorphism. Congressus Numerantium, 30 (1981), 45–87. 124 Magma Overview V2.18 (December 16, 2011)

[34] J. M¨uller-Quadeand T. Beth. Calculating Generators for Invariant Fields of Linear Algebraic Groups, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (Honolulu, HI, 1999), LNCS 1719, Springer, 1999, 392–403. [35] P. M. Neumann. Some algorithms for computing with ﬁnite permutation groups. Groups - St. Andrews 1985, London Math. Soc. Lecture Notes 121, 1986.

[36] A. Pizer. An algorithm for computing modular forms on Γ0(N), J. Algebra 64 (1980), no. 2, 340–390. [37] Miles Reid and Kaori Suzuki. Cascades of projections from log del Pezzo surfaces. In Number theory and algebraic geometry – to Peter Swinnerton-Dyer on his 75th birthday, pages 227– 249. Cambridge University Press, Cambridge, 2003. [38] C. M. Roney-Dougal and W. R. Unger. The aﬃne primitive permutation groups of degree less than 1000. J. Symb. Comput. 35 (2003), pp. 421–439.

[39] C. M. Roney-Dougal. The primitive groups of degree less than 2500. Submitted. [40] A. Steel. A New Algorithm for the Computation of Canonical Forms of Matrices over Fields, J. Symb. Comp., 24 (1997), 409–432. [41] A. Steel. A New Scheme for Computing with Algebraically Closed Fields, Proceedings of ANTS V, ed. C. Fieker and D.R. Kohel, Springer LCNS 2369 (2002), 493–507. [42] A. Steel. Computing With Algebraic Closures, J. Symb. Comput., 45(3): 342–372 (2010). [43] A. Steel. Conquering Inseparability: Primary Decomposition and Multivariate Factorization over Algebraic Function Fields of Positive Characteristic. JSC 40 (2005), 1053–1075. [44] A. Steel. Magma V2.12 is up to 2.3 times faster than GMP 4.1.4 for large integer multiplication, http://magma.maths.usyd.edu.au/users/allan/gb/. [45] A. Steel. Gr¨obnerBasis Timings Page, http://magma.maths.usyd.edu.au/users/allan/gb/. [46] W. A. Stein. Explicit approaches to the theory of modular abelian varieties. Ph.D. thesis, University of California, 2000. [47] W. A. Stein. Modular forms: a computational approach, with an appendix by Paul Gunnels. Amer. Math. Soc. Graduate Studies in Mathematics, Volume 79 (2007). [48] Robert J. Walker. Algebraic Curves, pages 98–99. Springer-Verlag, 1978.